(1)

結合律より、
$$P\vee (Q\vee R)\iff P\vee Q\vee R$$

(2)

$$P\vee (Q\wedge R)\iff (P\vee Q)\wedge (P\vee R)$$

(3)

$$\begin{array}{rl}P\vee (Q\to R)& \iff P\vee (\mathrm{\neg }Q\vee R)\\ & \iff P\vee \mathrm{\neg }Q\vee R\\ & \iff (P\leftarrow Q)\vee R\end{array}$$

(4)

$$\begin{array}{rl}P\vee (Q\leftarrow R)& \iff P\vee (Q\vee \mathrm{\neg }R)\\ & \iff P\vee Q\vee \mathrm{\neg }R\\ & \iff (P\vee Q)\leftarrow R\end{array}$$

(5)

$$\begin{array}{rl}P\vee (Q\leftrightarrow R)& \iff P\vee ((\mathrm{\neg }Q\vee R)\wedge (Q\vee \mathrm{\neg }R))\\ & \iff (P\vee \mathrm{\neg }Q\vee R)\wedge (P\vee Q\vee \mathrm{\neg }R)\\ & \iff (P\vee R\vee \mathrm{\neg }Q)\wedge (P\vee Q\vee \mathrm{\neg }R)\\ & \iff (P\vee R\vee (\mathrm{\neg }P\wedge \mathrm{\neg }Q))\wedge (P\vee Q\vee (\mathrm{\neg }P\wedge \mathrm{\neg }R))\\ & \iff ((P\vee R)\vee \mathrm{\neg }(P\vee Q))\wedge ((P\vee Q)\vee \mathrm{\neg }(P\vee R))\\ & \iff (P\vee Q)\leftrightarrow (P\vee R)\end{array}$$ $Q$を$\mathrm{\neg }Q$にして$R$を$\mathrm{\neg }R$にしても成り立つので、
$$\begin{array}{rl}P\vee (Q\leftrightarrow R)& \iff P\vee (\mathrm{\neg }Q\leftrightarrow \mathrm{\neg }R)\\ & \iff (P\vee \mathrm{\neg }Q)\leftrightarrow (P\vee \mathrm{\neg }R)\\ & \iff (P\leftarrow Q)\leftrightarrow (P\leftarrow R)\end{array}$$

(5)-2

$$\begin{array}{rl}P\vee (Q\leftrightarrow R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\wedge (Q\nleftrightarrow R)\}\\ & \iff \mathrm{\neg }(\mathrm{\neg }P\wedge \{(Q\wedge \mathrm{\neg }R)\vee (\mathrm{\neg }Q\wedge R)\})\\ & \iff \mathrm{\neg }\{(\mathrm{\neg }P\wedge (Q\wedge \mathrm{\neg }R))\vee (\mathrm{\neg }P\wedge (\mathrm{\neg }Q\wedge R))\}\\ & \iff \mathrm{\neg }\{((\mathrm{\neg }P\wedge Q)\wedge \mathrm{\neg }R)\vee (\mathrm{\neg }Q\wedge (\mathrm{\neg }P\wedge R))\}\\ & \iff \mathrm{\neg }\{((\mathrm{\neg }P\wedge Q)\wedge (P\vee \mathrm{\neg }R))\vee ((P\vee \mathrm{\neg }Q)\wedge (\mathrm{\neg }P\wedge R))\}\\ & \iff \mathrm{\neg }\{((P\nleftarrow Q)\wedge (P\leftarrow R))\vee ((P\leftarrow Q)\wedge (P\nleftarrow R))\}\\ & \iff \mathrm{\neg }\{((P\nleftarrow Q)\wedge \mathrm{\neg }(P\nleftarrow R))\vee (\mathrm{\neg }(P\nleftarrow Q)\wedge (P\nleftarrow R))\}\\ & \iff \mathrm{\neg }\{(P\nleftarrow Q)\nleftrightarrow (P\nleftarrow R)\}\\ & \iff (P\nleftarrow Q)\leftrightarrow (P\nleftarrow R)\\ & \iff (P\leftarrow Q)\leftrightarrow (P\leftarrow R)\end{array}$$

(6)

$$\begin{array}{rl}P\vee (Q\downarrow R)& \iff P\vee (\mathrm{\neg }Q\wedge \mathrm{\neg }R)\\ & \iff (P\vee \mathrm{\neg }Q)\wedge (P\vee \mathrm{\neg }R)\\ & \iff (P\leftarrow Q)\wedge (P\leftarrow R)\end{array}$$

(7)

$$\begin{array}{rl}P\vee (Q\uparrow R)& \iff P\vee (\mathrm{\neg }Q\vee \mathrm{\neg }R)\\ & \iff P\vee \mathrm{\neg }Q\vee \mathrm{\neg }R\\ & \iff (P\leftarrow Q)\leftarrow R\end{array}$$

(8)

$$\begin{array}{rl}P\vee (Q\nrightarrow R)& \iff P\vee (Q\wedge \mathrm{\neg }R)\\ & \iff (P\vee Q)\wedge (P\vee \mathrm{\neg }R)\\ & \iff (P\vee Q)\wedge (P\leftarrow R)\end{array}$$

(9)

$$\begin{array}{rl}P\vee (Q\nleftarrow R)& \iff P\vee (\mathrm{\neg }Q\wedge R)\\ & \iff (P\vee \mathrm{\neg }Q)\wedge (P\vee R)\\ & \iff (P\leftarrow Q)\wedge (P\vee R)\end{array}$$

(10)

(5)より、
$$\begin{array}{rl}P\vee (Q\nleftrightarrow R)& \iff P\vee (\mathrm{\neg }Q\leftrightarrow R)\\ & \iff (P\vee \mathrm{\neg }Q)\leftrightarrow (P\vee R)\\ & \iff (P\leftarrow Q)\leftrightarrow (P\vee R)\end{array}$$ $$\begin{array}{rl}P\vee (Q\nleftrightarrow R)& \iff P\vee (\mathrm{\neg }Q\nleftrightarrow \mathrm{\neg }R)\\ & \iff (P\leftarrow \mathrm{\neg }Q)\leftrightarrow (P\vee \mathrm{\neg }R)\\ & \iff (P\vee Q)\leftrightarrow (P\leftarrow R)\\ & \iff (P\vee Q)\nleftrightarrow (P\nleftarrow R)\end{array}$$

(11)

分配律より、
$$P\wedge (Q\vee R)\iff (P\wedge Q)\vee (P\wedge R)$$

(12)

結合律より、
$$P\wedge (Q\wedge R)\iff P\wedge Q\wedge R$$

(13)

$$\begin{array}{rl}P\wedge (Q\to R)& \iff P\wedge (\mathrm{\neg }Q\vee R)\\ & \iff (P\wedge \mathrm{\neg }Q)\vee (P\wedge R)\\ & \iff (P\nrightarrow Q)\vee (P\wedge R)\end{array}$$

(14)

$$\begin{array}{rl}P\wedge (Q\leftarrow R)& \iff P\wedge (Q\vee \mathrm{\neg }R)\\ & \iff (P\wedge Q)\vee (P\wedge \mathrm{\neg }R)\\ & \iff (P\wedge Q)\vee (P\nrightarrow R)\end{array}$$

(15)

(5)より、
$$\begin{array}{rl}P\wedge (Q\leftrightarrow R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\vee \mathrm{\neg }(Q\leftrightarrow R)\}\\ & \iff \mathrm{\neg }\{\mathrm{\neg }P\vee (\mathrm{\neg }Q\leftrightarrow R)\}\\ & \iff \mathrm{\neg }\{(\mathrm{\neg }P\vee \mathrm{\neg }Q)\leftrightarrow (\mathrm{\neg }P\vee R)\}\\ & \iff \mathrm{\neg }\{(P\uparrow Q)\leftrightarrow (P\to R)\}\\ & \iff (P\uparrow Q)\nleftrightarrow (P\to R)\\ & \iff (P\wedge Q)\leftrightarrow (P\to R)\end{array}$$ $$\begin{array}{rl}P\wedge (Q\leftrightarrow R)& \iff P\wedge (\mathrm{\neg }Q\leftrightarrow \mathrm{\neg }R)\\ & \iff (P\wedge \mathrm{\neg }Q)\leftrightarrow (P\to \mathrm{\neg }R)\\ & \iff (P\nrightarrow Q)\leftrightarrow (P\uparrow R)\\ & \iff (P\to Q)\leftrightarrow (P\wedge R)\end{array}$$

(16)

$$\begin{array}{rl}P\wedge (Q\downarrow R)& \iff P\wedge \mathrm{\neg }Q\wedge \mathrm{\neg }R\\ & \iff (P\nrightarrow Q)\nrightarrow R\end{array}$$

(17)

$$\begin{array}{rl}P\wedge (Q\uparrow R)& \iff P\wedge (\mathrm{\neg }Q\vee \mathrm{\neg }R)\\ & \iff (P\wedge \mathrm{\neg }Q)\vee (P\wedge \mathrm{\neg }R)\\ & \iff (P\nrightarrow Q)\vee (P\nrightarrow R)\end{array}$$

(18)

$$\begin{array}{rl}P\wedge (Q\nrightarrow R)& \iff P\wedge Q\wedge \mathrm{\neg }R\\ & \iff (P\wedge Q)\nrightarrow R\end{array}$$

(19)

$$\begin{array}{rl}P\wedge (Q\nleftarrow R)& \iff P\wedge \mathrm{\neg }Q\wedge R\\ & \iff (P\nrightarrow Q)\wedge R\end{array}$$

(20)

(5)より、
$$\begin{array}{rl}P\wedge (Q\nleftrightarrow R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\vee (Q\leftrightarrow R)\}\\ & \iff \mathrm{\neg }\{(\mathrm{\neg }P\vee Q)\leftrightarrow (\mathrm{\neg }P\vee R)\}\\ & \iff \mathrm{\neg }\{(P\to Q)\leftrightarrow (P\to R)\}\\ & \iff (P\to Q)\nleftrightarrow (P\to R)\end{array}$$ $$\begin{array}{rl}P\wedge (Q\nleftrightarrow R)& \iff P\wedge (\mathrm{\neg }Q\nleftrightarrow \mathrm{\neg }R)\\ & \iff (P\to \mathrm{\neg }Q)\nleftrightarrow (P\to \mathrm{\neg }R)\\ & \iff (P\uparrow Q)\nleftrightarrow (P\uparrow R)\\ & \iff (P\wedge Q)\nleftrightarrow (P\wedge R)\end{array}$$

(21)

$$\begin{array}{rl}P\to (Q\vee R)& \iff \mathrm{\neg }P\vee Q\vee R\\ & \iff (P\to Q)\vee R\end{array}$$

(22)

$$\begin{array}{rl}P\to (Q\wedge R)& \iff \mathrm{\neg }P\vee (Q\wedge R)\\ & \iff (\mathrm{\neg }P\vee Q)\wedge (\mathrm{\neg }P\vee R)\\ & \iff (P\to Q)\wedge (P\to R)\end{array}$$

(23)

$$\begin{array}{rl}P\to (Q\to R)& \iff \mathrm{\neg }P\vee \mathrm{\neg }Q\vee R\\ & \iff (P\uparrow Q)\vee R\end{array}$$

(24)

$$\begin{array}{rl}P\to (Q\leftarrow R)& \iff \mathrm{\neg }P\vee Q\vee \mathrm{\neg }R\\ & \iff (P\to Q)\leftarrow R\end{array}$$

(25)

(5)より、
$$\begin{array}{rl}P\to (Q\leftrightarrow R)& \iff \mathrm{\neg }P\vee (Q\leftrightarrow R)\\ & \iff (\mathrm{\neg }P\vee Q)\leftrightarrow (\mathrm{\neg }P\vee R)\\ & \iff (P\to Q)\leftrightarrow (P\to R)\end{array}$$ $$\begin{array}{rl}P\to (Q\leftrightarrow R)& \iff P\to (\mathrm{\neg }Q\leftrightarrow \mathrm{\neg }R)\\ & \iff (P\to \mathrm{\neg }Q)\leftrightarrow (P\to \mathrm{\neg }R)\\ & \iff (P\uparrow Q)\leftrightarrow (P\uparrow R)\end{array}$$

(26)

$$\begin{array}{rl}P\to (Q\downarrow R)& \iff \mathrm{\neg }P\vee (\mathrm{\neg }Q\wedge \mathrm{\neg }R)\\ & \iff (\mathrm{\neg }P\vee \mathrm{\neg }Q)\wedge (\mathrm{\neg }P\vee \mathrm{\neg }R)\\ & \iff (P\uparrow Q)\wedge (P\uparrow R)\end{array}$$

(27)

$$\begin{array}{rl}P\to (Q\uparrow R)& \iff \mathrm{\neg }P\vee \mathrm{\neg }Q\vee \mathrm{\neg }R\\ & \iff (P\uparrow Q)\leftarrow R\end{array}$$

(28)

$$\begin{array}{rl}P\to (Q\nrightarrow R)& \iff \mathrm{\neg }P\vee (Q\wedge \mathrm{\neg }R)\\ & \iff (\mathrm{\neg }P\vee Q)\wedge (\mathrm{\neg }P\vee \mathrm{\neg }R)\\ & \iff (P\to Q)\wedge (P\uparrow R)\end{array}$$

(29)

$$\begin{array}{rl}P\to (Q\nleftarrow R)& \iff \mathrm{\neg }P\vee (\mathrm{\neg }Q\wedge R)\\ & \iff (\mathrm{\neg }P\vee \mathrm{\neg }Q)\wedge (\mathrm{\neg }P\vee R)\\ & \iff (P\uparrow Q)\wedge (P\to R)\end{array}$$

(30)

(5)より、
$$\begin{array}{rl}P\to (Q\nleftrightarrow R)& \iff \mathrm{\neg }P\vee (\mathrm{\neg }Q\leftrightarrow R)\\ & \iff (\mathrm{\neg }P\vee \mathrm{\neg }Q)\leftrightarrow (\mathrm{\neg }P\vee R)\\ & \iff (P\uparrow Q)\leftrightarrow (P\to R)\\ & \iff (P\wedge Q)\nleftrightarrow (P\to R)\end{array}$$ $$\begin{array}{rl}P\to (Q\nleftrightarrow R)& \iff P\to (\mathrm{\neg }Q\nleftrightarrow \mathrm{\neg }R)\\ & \iff (P\wedge \mathrm{\neg }Q)\nleftrightarrow (P\to \mathrm{\neg }R)\\ & \iff (P\nrightarrow Q)\nleftrightarrow (P\uparrow R)\\ & \iff (P\to Q)\nleftrightarrow (P\wedge R)\end{array}$$

(31)

$$\begin{array}{rl}P\leftarrow (Q\vee R)& \iff P\vee \mathrm{\neg }(Q\vee R)\\ & \iff P\vee (\mathrm{\neg }Q\wedge \mathrm{\neg }R)\\ & \iff (P\vee \mathrm{\neg }Q)\wedge (P\vee \mathrm{\neg }R)\\ & \iff (P\leftarrow Q)\wedge (P\leftarrow R)\end{array}$$

(32)

$$\begin{array}{rl}P\leftarrow (Q\wedge R)& \iff P\vee \mathrm{\neg }(Q\wedge R)\\ & \iff P\vee (\mathrm{\neg }Q\vee \mathrm{\neg }R)\\ & \iff P\vee \mathrm{\neg }Q\vee \mathrm{\neg }R\\ & \iff (P\leftarrow Q)\vee \mathrm{\neg }R\\ & \iff (P\leftarrow Q)\leftarrow R\end{array}$$

(33)

$$\begin{array}{rl}P\leftarrow (Q\to R)& \iff P\vee \mathrm{\neg }(\mathrm{\neg }Q\vee R)\\ & \iff P\vee (Q\wedge \mathrm{\neg }R)\\ & \iff (P\vee Q)\wedge (P\vee \mathrm{\neg }R)\\ & \iff (P\vee Q)\wedge (P\leftarrow R)\end{array}$$

(34)

$$\begin{array}{rl}P\leftarrow (Q\leftarrow R)& \iff P\vee \mathrm{\neg }(Q\vee \mathrm{\neg }R)\\ & \iff P\vee (\mathrm{\neg }Q\wedge R)\\ & \iff (P\vee \mathrm{\neg }Q)\wedge (P\vee R)\\ & \iff (P\leftarrow Q)\wedge (P\vee R)\end{array}$$

(35)

(5)より、
$$\begin{array}{rl}P\leftarrow (Q\leftrightarrow R)& \iff P\vee \mathrm{\neg }(Q\leftrightarrow R)\\ & \iff P\vee (\mathrm{\neg }Q\leftrightarrow R)\\ & \iff (P\vee \mathrm{\neg }Q)\leftrightarrow (P\vee R)\\ & \iff (P\leftarrow Q)\leftrightarrow (P\vee R)\end{array}$$ $$\begin{array}{rl}P\leftarrow (Q\leftrightarrow R)& \iff P\leftarrow (\mathrm{\neg }Q\leftrightarrow \mathrm{\neg }R)\\ & \iff (P\leftarrow \mathrm{\neg }Q)\leftrightarrow (P\vee \mathrm{\neg }R)\\ & \iff (P\vee Q)\leftrightarrow (P\leftarrow R)\end{array}$$

(36)

$$\begin{array}{rl}P\leftarrow (Q\downarrow R)& \iff P\vee \mathrm{\neg }(Q\downarrow R)\\ & \iff P\vee (Q\vee R)\\ & \iff (P\vee Q)\vee R\end{array}$$

(37)

$$\begin{array}{rl}P\leftarrow (Q\uparrow R)& \iff P\vee \mathrm{\neg }(Q\uparrow R)\\ & \iff P\vee (Q\wedge R)\\ & \iff (P\vee Q)\wedge (P\vee R)\end{array}$$

(38)

$$\begin{array}{rl}P\leftarrow (Q\nrightarrow R)& \iff P\vee \mathrm{\neg }(Q\nrightarrow R)\\ & \iff P\vee (Q\to R)\\ & \iff P\vee (\mathrm{\neg }Q\vee R)\\ & \iff (P\vee \mathrm{\neg }Q)\vee R\\ & \iff (P\leftarrow Q)\vee R\end{array}$$

(39)

$$\begin{array}{rl}P\leftarrow (Q\nleftarrow R)& \iff P\vee \mathrm{\neg }(Q\nleftarrow R)\\ & \iff P\vee (Q\leftarrow R)\\ & \iff P\vee (Q\vee \mathrm{\neg }R)\\ & \iff (P\vee Q)\vee \mathrm{\neg }R\\ & \iff (P\vee Q)\leftarrow R\end{array}$$

(40)

(5)より、
$$\begin{array}{rl}P\leftarrow (Q\nleftrightarrow R)& \iff P\vee \mathrm{\neg }(Q\nleftrightarrow R)\\ & \iff P\vee (Q\leftrightarrow R)\\ & \iff (P\vee Q)\leftrightarrow (P\vee R)\end{array}$$ $$\begin{array}{rl}P\leftarrow (Q\nleftrightarrow R)& \iff P\leftarrow (\mathrm{\neg }Q\nleftrightarrow \mathrm{\neg }R)\\ & \iff (P\vee \mathrm{\neg }Q)\leftrightarrow (P\vee \mathrm{\neg }R)\\ & \iff (P\leftarrow Q)\leftrightarrow (P\leftarrow R)\end{array}$$

(41)

(21)(31)より、
$$\begin{array}{rl}P\leftrightarrow (Q\vee R)& \iff (P\to (Q\vee R))\wedge (P\leftarrow (Q\vee R))\\ & \iff \{(P\to Q)\vee R\}\wedge \{(P\leftarrow Q)\wedge (P\leftarrow R)\}\\ & \iff \{(P\to Q)\wedge (P\leftarrow Q)\wedge (P\leftarrow R)\}\vee \{R\wedge (P\leftarrow Q)\wedge (P\leftarrow R)\}\\ & \iff \{(P\leftrightarrow Q)\wedge (P\leftarrow R)\}\vee \{R\wedge P\wedge (P\leftarrow Q)\}\\ & \iff \{(P\leftrightarrow Q)\wedge (P\leftarrow R)\}\vee (P\wedge R)\end{array}$$ $$\begin{array}{rl}P\leftrightarrow (Q\vee R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\leftrightarrow (Q\vee R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\leftrightarrow Q)\wedge (\mathrm{\neg }P\leftarrow R)\}\vee (\mathrm{\neg }P\wedge R)\}\\ & \iff \mathrm{\neg }\{\{(P\nleftrightarrow Q)\wedge (P\uparrow R)\}\vee (P\nleftarrow R)\}\\ & \iff \{(P\leftrightarrow Q)\vee (P\wedge R)\}\wedge (P\leftarrow R)\end{array}$$

(41)-2

直接導出する場合は、分配法則を使えば出せる。
$$\begin{array}{rl}P\leftrightarrow (Q\vee R)& \iff \{(P\leftrightarrow Q)\wedge (P\leftarrow R)\}\vee (P\wedge R)\\ & \iff \{(P\leftrightarrow Q)\vee (P\wedge R)\}\wedge \{(P\leftarrow R)\vee (P\wedge R)\}\\ & \iff \{(P\leftrightarrow Q)\vee (P\wedge R)\}\wedge \{\mathrm{\neg }R\vee P\vee (P\wedge R)\}\\ & \iff \{(P\leftrightarrow Q)\vee (P\wedge R)\}\wedge (\mathrm{\neg }R\vee P)\\ & \iff \{(P\leftrightarrow Q)\vee (P\wedge R)\}\wedge (P\leftarrow R)\end{array}$$

(42)

(41)より、
$$\begin{array}{rl}P\leftrightarrow (Q\wedge R)& \iff \mathrm{\neg }P\leftrightarrow \mathrm{\neg }(Q\wedge R)\\ & \iff \mathrm{\neg }P\leftrightarrow (\mathrm{\neg }Q\vee \mathrm{\neg }R)\\ & \iff \{(\mathrm{\neg }P\leftrightarrow \mathrm{\neg }Q)\vee (\mathrm{\neg }P\wedge \mathrm{\neg }R)\}\wedge (\mathrm{\neg }P\leftarrow \mathrm{\neg }R)\\ & \iff \{(P\leftrightarrow Q)\vee (P\downarrow R)\}\wedge (P\to R)\end{array}$$ $$\begin{array}{rl}P\leftrightarrow (Q\wedge R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\leftrightarrow (Q\wedge R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\leftrightarrow Q)\vee (\mathrm{\neg }P\downarrow R)\}\wedge (\mathrm{\neg }P\to R)\}\\ & \iff \mathrm{\neg }\{\{(P\nleftrightarrow Q)\vee (P\nrightarrow R)\}\wedge (P\vee R)\}\\ & \iff \{(P\leftrightarrow Q)\wedge (P\to R)\}\vee (P\downarrow R)\end{array}$$

(43)

(41)より、
$$\begin{array}{rl}P\leftrightarrow (Q\to R)& \iff P\leftrightarrow (\mathrm{\neg }Q\vee R)\\ & \iff \{(P\leftrightarrow \mathrm{\neg }Q)\vee (P\wedge R)\}\wedge (P\leftarrow R)\\ & \iff \{(P\nleftrightarrow Q)\vee (P\wedge R)\}\wedge (P\leftarrow R)\\ & \iff \{(P\leftrightarrow Q)\uparrow (P\uparrow R)\}\wedge (P\leftarrow R)\end{array}$$ $$\begin{array}{rl}P\leftrightarrow (Q\to R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\leftrightarrow (Q\to R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\leftrightarrow Q)\uparrow (\mathrm{\neg }P\uparrow R)\}\wedge (\mathrm{\neg }P\leftarrow R)\}\\ & \iff \mathrm{\neg }\{\{(P\nleftrightarrow Q)\uparrow (P\leftarrow R)\}\wedge (P\uparrow R)\}\\ & \iff \{(P\nleftrightarrow Q)\wedge (P\leftarrow R)\}\vee (P\wedge R)\\ & \iff \{(P\leftrightarrow Q)\downarrow (P\nleftarrow R)\}\vee (P\wedge R)\end{array}$$

(44)

(41)より、
$$\begin{array}{rl}P\leftrightarrow (Q\leftarrow R)& \iff P\leftrightarrow (Q\vee \mathrm{\neg }R)\\ & \iff \{(P\leftrightarrow Q)\vee (P\wedge \mathrm{\neg }R)\}\wedge (P\leftarrow \mathrm{\neg }R)\\ & \iff \{(P\leftrightarrow Q)\vee (P\nrightarrow R)\}\wedge (P\vee R)\end{array}$$ $$\begin{array}{rl}P\leftrightarrow (Q\leftarrow R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\leftrightarrow (Q\leftarrow R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\leftrightarrow Q)\vee (\mathrm{\neg }P\nrightarrow R)\}\wedge (\mathrm{\neg }P\vee R)\}\\ & \iff \mathrm{\neg }\{\{(P\nleftrightarrow Q)\vee (P\downarrow R)\}\wedge (P\to R)\}\\ & \iff \{(P\leftrightarrow Q)\wedge (P\vee R)\}\vee (P\nrightarrow R)\end{array}$$

(45)

$$\begin{array}{rl}P\leftrightarrow (Q\leftrightarrow R)& \iff \{P\wedge (Q\leftrightarrow R)\}\vee \{\mathrm{\neg }P\wedge \mathrm{\neg }(Q\leftrightarrow R)\}\\ & \iff \{P\wedge \{(Q\wedge R)\vee (\mathrm{\neg }Q\wedge \mathrm{\neg }R)\}\}\vee \{\mathrm{\neg }P\wedge \mathrm{\neg }\{(Q\wedge R)\vee (\mathrm{\neg }Q\wedge \mathrm{\neg }R)\}\}\\ & \iff \{P\wedge \{(Q\wedge R)\vee (\mathrm{\neg }Q\wedge \mathrm{\neg }R)\}\}\vee \{\mathrm{\neg }P\wedge \{(\mathrm{\neg }Q\vee \mathrm{\neg }R)\wedge (Q\vee R)\}\}\\ & \iff \{P\wedge \{(Q\wedge R)\vee (\mathrm{\neg }Q\wedge \mathrm{\neg }R)\}\}\vee \{\mathrm{\neg }P\wedge \{(\mathrm{\neg }Q\wedge R)\vee (\mathrm{\neg }R\wedge Q)\}\}\\ & \iff (P\wedge Q\wedge R)\vee (P\wedge \mathrm{\neg }Q\wedge \mathrm{\neg }R)\vee (\mathrm{\neg }P\wedge \mathrm{\neg }Q\wedge R)\vee (\mathrm{\neg }P\wedge Q\wedge \mathrm{\neg }R)\\ & \iff \{\{(P\wedge Q)\vee (\mathrm{\neg }P\wedge \mathrm{\neg }Q)\}\wedge R\}\vee \{\{(P\wedge \mathrm{\neg }Q)\vee (\mathrm{\neg }P\wedge Q)\}\wedge \mathrm{\neg }R\}\\ & \iff \{(P\leftrightarrow Q)\wedge R\}\vee \{(P\nleftrightarrow Q)\wedge \mathrm{\neg }R\}\\ & \iff \{(P\leftrightarrow Q)\wedge R\}\vee \{\mathrm{\neg }(P\leftrightarrow Q)\wedge \mathrm{\neg }R\}\\ & \iff (P\leftrightarrow Q)\leftrightarrow R\end{array}$$

(46)

(41)より、
$$\begin{array}{rl}P\leftrightarrow (Q\downarrow R)& \iff \mathrm{\neg }P\leftrightarrow (Q\vee R)\\ & \iff \{(\mathrm{\neg }P\leftrightarrow Q)\vee (\mathrm{\neg }P\wedge R)\}\wedge (\mathrm{\neg }P\leftarrow R)\\ & \iff \{(P\nleftrightarrow Q)\vee (P\nleftarrow R)\}\wedge (P\uparrow R)\\ & \iff \{(P\leftrightarrow Q)\uparrow (P\leftarrow R)\}\wedge (P\uparrow R)\end{array}$$ $$\begin{array}{rl}P\leftrightarrow (Q\downarrow R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\leftrightarrow (Q\downarrow R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\leftrightarrow Q)\uparrow (\mathrm{\neg }P\leftarrow R)\}\wedge (\mathrm{\neg }P\uparrow R)\}\\ & \iff \mathrm{\neg }\{\{(P\nleftrightarrow Q)\uparrow (P\uparrow R)\}\wedge (P\leftarrow R)\}\\ & \iff \{(P\nleftrightarrow Q)\wedge (P\uparrow R)\}\vee (P\nleftarrow R)\\ & \iff \{(P\leftrightarrow Q)\downarrow (P\wedge R)\}\vee (P\nleftarrow R)\end{array}$$

(47)

(41)より、
$$\begin{array}{rl}P\leftrightarrow (Q\uparrow R)& \iff P\leftrightarrow \mathrm{\neg }(Q\wedge R)\\ & \iff P\leftrightarrow (\mathrm{\neg }Q\vee \mathrm{\neg }R)\\ & \iff \{(P\leftrightarrow \mathrm{\neg }Q)\vee (P\wedge \mathrm{\neg }R)\}\wedge (P\leftarrow \mathrm{\neg }R)\\ & \iff \{(P\nleftrightarrow Q)\vee (P\nrightarrow R)\}\wedge (P\vee R)\\ & \iff \{(P\leftrightarrow Q)\uparrow (P\to R)\}\wedge (P\vee R)\end{array}$$ $$\begin{array}{rl}P\leftrightarrow (Q\uparrow R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\leftrightarrow (Q\uparrow R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\leftrightarrow Q)\uparrow (\mathrm{\neg }P\to R)\}\wedge (\mathrm{\neg }P\vee R)\}\\ & \iff \mathrm{\neg }\{\{(P\nleftrightarrow Q)\uparrow (P\vee R)\}\wedge (P\to R)\}\\ & \iff \{(P\nleftrightarrow Q)\wedge (P\vee R)\}\vee (P\nrightarrow R)\\ & \iff \{(P\leftrightarrow Q)\downarrow (P\downarrow R)\}\vee (P\nrightarrow R)\end{array}$$

(48)

(41)より、
$$\begin{array}{rl}P\leftrightarrow (Q\nrightarrow R)& \iff \mathrm{\neg }P\leftrightarrow (Q\to R)\\ & \iff \mathrm{\neg }P\leftrightarrow (\mathrm{\neg }Q\vee R)\\ & \iff \{(\mathrm{\neg }P\leftrightarrow \mathrm{\neg }Q)\vee (\mathrm{\neg }P\wedge R)\}\wedge (\mathrm{\neg }P\leftarrow R)\\ & \iff \{(P\leftrightarrow Q)\vee (P\nleftarrow R)\}\wedge (P\uparrow R)\end{array}$$ $$\begin{array}{rl}P\leftrightarrow (Q\nrightarrow R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\leftrightarrow (Q\nrightarrow R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\leftrightarrow Q)\vee (\mathrm{\neg }P\nleftarrow R)\}\wedge (\mathrm{\neg }P\uparrow R)\}\\ & \iff \mathrm{\neg }\{\{(P\nleftrightarrow Q)\vee (P\wedge R)\}\wedge (P\leftarrow R)\}\\ & \iff \{(P\leftrightarrow Q)\wedge (P\uparrow R)\}\vee (P\nleftarrow R)\end{array}$$

(49)

(41)より、
$$\begin{array}{rl}P\leftrightarrow (Q\nleftarrow R)& \iff \mathrm{\neg }P\leftrightarrow (Q\leftarrow R)\\ & \iff \mathrm{\neg }P\leftrightarrow (Q\vee \mathrm{\neg }R)\\ & \iff \{(\mathrm{\neg }P\leftrightarrow Q)\vee (\mathrm{\neg }P\wedge \mathrm{\neg }R)\}\wedge (\mathrm{\neg }P\leftarrow \mathrm{\neg }R)\\ & \iff \{(P\nleftrightarrow Q)\vee (P\downarrow R)\}\wedge (P\to R)\\ & \iff \{(P\leftrightarrow Q)\uparrow (P\vee R)\}\wedge (P\to R)\end{array}$$ $$\begin{array}{rl}P\leftrightarrow (Q\nleftarrow R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\leftrightarrow (Q\nleftarrow R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\leftrightarrow Q)\uparrow (\mathrm{\neg }P\vee R)\}\wedge (\mathrm{\neg }P\to R)\}\\ & \iff \mathrm{\neg }\{\{(P\nleftrightarrow Q)\uparrow (P\to R)\}\wedge (P\vee R)\}\\ & \iff \{\{(P\nleftrightarrow Q)\wedge (P\to R)\}\vee (P\downarrow R)\}\\ & \iff \{\{(P\leftrightarrow Q)\downarrow (P\nrightarrow R)\}\vee (P\downarrow R)\}\end{array}$$

(50)

(45)より、
$$\begin{array}{rl}P\leftrightarrow (Q\nleftrightarrow R)& \iff P\leftrightarrow (Q\leftrightarrow \mathrm{\neg }R)\\ & \iff (P\leftrightarrow Q)\leftrightarrow \mathrm{\neg }R\\ & \iff (P\leftrightarrow Q)\nleftrightarrow R\end{array}$$

(51)

$$\begin{array}{rl}P\downarrow (Q\vee R)& \iff \mathrm{\neg }P\wedge \mathrm{\neg }(Q\vee R)\\ & \iff \mathrm{\neg }P\wedge \mathrm{\neg }Q\wedge \mathrm{\neg }R\\ & \iff (P\downarrow Q)\nrightarrow R\end{array}$$

(52)

$$\begin{array}{rl}P\downarrow (Q\wedge R)& \iff \mathrm{\neg }P\wedge \mathrm{\neg }(Q\wedge R)\\ & \iff \mathrm{\neg }P\wedge (\mathrm{\neg }Q\vee \mathrm{\neg }R)\\ & \iff (\mathrm{\neg }P\wedge \mathrm{\neg }Q)\vee (\mathrm{\neg }P\wedge \mathrm{\neg }R)\\ & \iff (P\downarrow Q)\vee (P\downarrow R)\end{array}$$

(53)

$$\begin{array}{rl}P\downarrow (Q\to R)& \iff \mathrm{\neg }P\wedge \mathrm{\neg }(Q\to R)\\ & \iff \mathrm{\neg }P\wedge Q\wedge \mathrm{\neg }R\\ & \iff (P\nleftarrow Q)\nrightarrow R\\ & \iff (P\leftarrow Q)\nleftarrow \mathrm{\neg }R\\ & \iff (P\leftarrow Q)\downarrow R\end{array}$$

(54)

$$\begin{array}{rl}P\downarrow (Q\leftarrow R)& \iff \mathrm{\neg }P\wedge \mathrm{\neg }(Q\leftarrow R)\\ & \iff \mathrm{\neg }P\wedge \mathrm{\neg }Q\wedge R\\ & \iff (P\downarrow Q)\wedge R\end{array}$$

(55)

(5)より、
$$\begin{array}{rl}P\downarrow (Q\leftrightarrow R)& \iff \mathrm{\neg }P\wedge (Q\nleftrightarrow R)\\ & \iff \mathrm{\neg }\{P\vee (Q\leftrightarrow R)\}\\ & \iff \mathrm{\neg }\{(P\vee Q)\leftrightarrow (P\vee R)\}\\ & \iff (P\vee Q)\nleftrightarrow (P\vee R)\\ & \iff (P\downarrow Q)\leftrightarrow (P\vee R)\end{array}$$ $$\begin{array}{rl}P\downarrow (Q\leftrightarrow R)& \iff P\downarrow (\mathrm{\neg }Q\leftrightarrow \mathrm{\neg }R)\\ & \iff (P\downarrow \mathrm{\neg }Q)\leftrightarrow (P\vee \mathrm{\neg }R)\\ & \iff (P\nleftarrow Q)\leftrightarrow (P\leftarrow R)\\ & \iff (P\leftarrow Q)\nleftrightarrow (P\leftarrow R)\end{array}$$

(56)

$$\begin{array}{rl}P\downarrow (Q\downarrow R)& \iff \mathrm{\neg }P\wedge \mathrm{\neg }(Q\downarrow R)\\ & \iff \mathrm{\neg }P\wedge (Q\vee R)\\ & \iff (\mathrm{\neg }P\wedge Q)\vee (\mathrm{\neg }P\wedge R)\\ & \iff (P\nleftarrow Q)\vee (P\nleftarrow R)\end{array}$$

(57)

$$\begin{array}{rl}P\downarrow (Q\uparrow R)& \iff \mathrm{\neg }P\wedge \mathrm{\neg }(Q\uparrow R)\\ & \iff \mathrm{\neg }P\wedge Q\wedge R\\ & \iff (P\nleftarrow Q)\wedge R\end{array}$$

(58)

$$\begin{array}{rl}P\downarrow (Q\nrightarrow R)& \iff \mathrm{\neg }P\wedge \mathrm{\neg }(Q\nrightarrow R)\\ & \iff \mathrm{\neg }P\wedge (\mathrm{\neg }Q\vee R)\\ & \iff (\mathrm{\neg }P\wedge \mathrm{\neg }Q)\vee (\mathrm{\neg }P\wedge R)\\ & \iff (P\downarrow Q)\vee (P\nleftarrow R)\end{array}$$

(59)

$$\begin{array}{rl}P\downarrow (Q\nleftarrow R)& \iff \mathrm{\neg }P\wedge \mathrm{\neg }(Q\nleftarrow R)\\ & \iff \mathrm{\neg }P\wedge (Q\vee \mathrm{\neg }R)\\ & \iff (\mathrm{\neg }P\wedge Q)\vee (\mathrm{\neg }P\wedge \mathrm{\neg }R)\\ & \iff (P\nleftarrow Q)\vee (P\downarrow R)\end{array}$$

(60)

(5)より、
$$\begin{array}{rl}P\downarrow (Q\nleftrightarrow R)& \iff \mathrm{\neg }P\wedge (Q\leftrightarrow R)\\ & \iff \mathrm{\neg }\{P\vee (\mathrm{\neg }Q\leftrightarrow R)\}\\ & \iff \mathrm{\neg }\{(P\vee \mathrm{\neg }Q)\leftrightarrow (P\vee R)\}\\ & \iff (P\leftarrow Q)\nleftrightarrow (P\vee R)\end{array}$$ $$\begin{array}{rl}P\downarrow (Q\nleftrightarrow R)& \iff P\downarrow (\mathrm{\neg }Q\nleftrightarrow \mathrm{\neg }R)\\ & \iff (P\leftarrow \mathrm{\neg }Q)\nleftrightarrow (P\vee \mathrm{\neg }R)\\ & \iff (P\vee Q)\nleftrightarrow (P\leftarrow R)\\ & \iff (P\downarrow Q)\nleftrightarrow (P\nleftarrow R)\end{array}$$

(61)

$$\begin{array}{rl}P\uparrow (Q\vee R)& \iff \mathrm{\neg }P\vee \mathrm{\neg }(Q\vee R)\\ & \iff \mathrm{\neg }P\vee (\mathrm{\neg }Q\wedge \mathrm{\neg }R)\\ & \iff (\mathrm{\neg }P\vee \mathrm{\neg }Q)\wedge (\mathrm{\neg }P\vee \mathrm{\neg }R)\\ & \iff (P\uparrow Q)\wedge (P\uparrow R)\end{array}$$

(62)

$$\begin{array}{rl}P\uparrow (Q\wedge R)& \iff \mathrm{\neg }P\vee \mathrm{\neg }(Q\wedge R)\\ & \iff \mathrm{\neg }P\vee (\mathrm{\neg }Q\vee \mathrm{\neg }R)\\ & \iff (\mathrm{\neg }P\vee \mathrm{\neg }Q)\vee \mathrm{\neg }R\\ & \iff (P\uparrow Q)\leftarrow R\end{array}$$

(63)

$$\begin{array}{rl}P\uparrow (Q\to R)& \iff \mathrm{\neg }P\vee \mathrm{\neg }(\mathrm{\neg }Q\vee R)\\ & \iff \mathrm{\neg }P\vee (Q\wedge \mathrm{\neg }R)\\ & \iff (\mathrm{\neg }P\vee Q)\wedge (\mathrm{\neg }P\vee \mathrm{\neg }R)\\ & \iff (P\to Q)\wedge (P\uparrow R)\end{array}$$

(64)

$$\begin{array}{rl}P\uparrow (Q\leftarrow R)& \iff \mathrm{\neg }P\vee \mathrm{\neg }(Q\vee \mathrm{\neg }R)\\ & \iff \mathrm{\neg }P\vee (\mathrm{\neg }Q\wedge R)\\ & \iff (\mathrm{\neg }P\vee \mathrm{\neg }Q)\wedge (\mathrm{\neg }P\vee R)\\ & \iff (P\uparrow Q)\wedge (P\to R)\end{array}$$

(65)

(5)より、
$$\begin{array}{rl}P\uparrow (Q\leftrightarrow R)& \iff \mathrm{\neg }P\vee (Q\nleftrightarrow R)\\ & \iff \mathrm{\neg }P\vee (\mathrm{\neg }Q\leftrightarrow R)\\ & \iff (\mathrm{\neg }P\vee \mathrm{\neg }Q)\leftrightarrow (\mathrm{\neg }P\vee R)\\ & \iff (P\uparrow Q)\leftrightarrow (P\to R)\end{array}$$ $$\begin{array}{rl}P\uparrow (Q\leftrightarrow R)& \iff P\uparrow (\mathrm{\neg }Q\leftrightarrow \mathrm{\neg }R)\\ & \iff (P\uparrow \mathrm{\neg }Q)\leftrightarrow (P\to \mathrm{\neg }R)\\ & \iff (P\to Q)\leftrightarrow (P\uparrow R)\end{array}$$

(66)

$$\begin{array}{rl}P\uparrow (Q\downarrow R)& \iff \mathrm{\neg }P\vee \mathrm{\neg }(Q\downarrow R)\\ & \iff \mathrm{\neg }P\vee (Q\vee R)\\ & \iff (\mathrm{\neg }P\vee Q)\vee R\\ & \iff (P\to Q)\vee R\end{array}$$

(67)

$$\begin{array}{rl}P\uparrow (Q\uparrow R)& \iff \mathrm{\neg }P\vee \mathrm{\neg }(Q\uparrow R)\\ & \iff \mathrm{\neg }P\vee (Q\wedge R)\\ & \iff (\mathrm{\neg }P\vee Q)\wedge (\mathrm{\neg }P\vee R)\\ & \iff (P\to Q)\wedge (P\to R)\end{array}$$

(68)

$$\begin{array}{rl}P\uparrow (Q\nrightarrow R)& \iff \mathrm{\neg }P\vee \mathrm{\neg }(Q\wedge \mathrm{\neg }R)\\ & \iff \mathrm{\neg }P\vee (\mathrm{\neg }Q\vee R)\\ & \iff (\mathrm{\neg }P\vee \mathrm{\neg }Q)\vee R\\ & \iff (P\uparrow Q)\vee R\end{array}$$

(69)

$$\begin{array}{rl}P\uparrow (Q\nleftarrow R)& \iff \mathrm{\neg }P\vee \mathrm{\neg }(\mathrm{\neg }Q\wedge R)\\ & \iff \mathrm{\neg }P\vee (Q\vee \mathrm{\neg }R)\\ & \iff (\mathrm{\neg }P\vee Q)\vee \mathrm{\neg }R\\ & \iff (P\to Q)\leftarrow R\end{array}$$

(70)

(5)より、
$$\begin{array}{rl}P\uparrow (Q\nleftrightarrow R)& \iff \mathrm{\neg }P\vee \mathrm{\neg }(Q\nleftrightarrow R)\\ & \iff \mathrm{\neg }P\vee (Q\leftrightarrow R)\\ & \iff (\mathrm{\neg }P\vee Q)\leftrightarrow (\mathrm{\neg }P\vee R)\\ & \iff (P\to Q)\leftrightarrow (P\to R)\end{array}$$ $$\begin{array}{rl}P\uparrow (Q\nleftrightarrow R)& \iff P\uparrow (\mathrm{\neg }Q\nleftrightarrow \mathrm{\neg }R)\\ & \iff (P\to \mathrm{\neg }Q)\leftrightarrow (P\to \mathrm{\neg }R)\\ & \iff (P\uparrow Q)\leftrightarrow (P\uparrow R)\\ & \iff (P\uparrow Q)\nleftrightarrow (P\wedge R)\end{array}$$

(71)

$$\begin{array}{rl}P\nrightarrow (Q\vee R)& \iff P\wedge \mathrm{\neg }(Q\vee R)\\ & \iff P\wedge (\mathrm{\neg }Q\wedge \mathrm{\neg }R)\\ & \iff (P\wedge \mathrm{\neg }Q)\wedge \mathrm{\neg }R\\ & \iff (P\nrightarrow Q)\nrightarrow R\end{array}$$

(72)

$$\begin{array}{rl}P\nrightarrow (Q\wedge R)& \iff P\wedge \mathrm{\neg }(Q\wedge R)\\ & \iff P\wedge (\mathrm{\neg }Q\vee \mathrm{\neg }R)\\ & \iff (P\wedge \mathrm{\neg }Q)\vee (P\wedge \mathrm{\neg }R)\\ & \iff (P\nrightarrow Q)\vee (P\nrightarrow R)\end{array}$$

(73)

$$\begin{array}{rl}P\nrightarrow (Q\to R)& \iff P\wedge \mathrm{\neg }(\mathrm{\neg }Q\vee R)\\ & \iff P\wedge (Q\wedge \mathrm{\neg }R)\\ & \iff (P\wedge Q)\wedge \mathrm{\neg }R\\ & \iff (P\wedge Q)\nrightarrow R\end{array}$$

(74)

$$\begin{array}{rl}P\nrightarrow (Q\leftarrow R)& \iff P\wedge \mathrm{\neg }(Q\vee \mathrm{\neg }R)\\ & \iff P\wedge (\mathrm{\neg }Q\wedge R)\\ & \iff (P\wedge \mathrm{\neg }Q)\wedge R\\ & \iff (P\nrightarrow Q)\wedge R\end{array}$$

(75)

(5)より、
$$\begin{array}{rl}P\nrightarrow (Q\leftrightarrow R)& \iff P\wedge \mathrm{\neg }(Q\leftrightarrow R)\\ & \iff \mathrm{\neg }\{\mathrm{\neg }P\vee (Q\leftrightarrow R)\}\\ & \iff \mathrm{\neg }\{(\mathrm{\neg }P\vee Q)\leftrightarrow (\mathrm{\neg }P\vee R)\}\\ & \iff (P\to Q)\nleftrightarrow (P\to R)\end{array}$$ $$\begin{array}{rl}P\nrightarrow (Q\leftrightarrow R)& \iff P\nrightarrow (\mathrm{\neg }Q\leftrightarrow \mathrm{\neg }R)\\ & \iff (P\to \mathrm{\neg }Q)\nleftrightarrow (P\to \mathrm{\neg }R)\\ & \iff (P\uparrow Q)\nleftrightarrow (P\uparrow R)\end{array}$$

(76)

$$\begin{array}{rl}P\nrightarrow (Q\downarrow R)& \iff P\wedge \mathrm{\neg }(Q\downarrow R)\\ & \iff P\wedge (Q\vee R)\\ & \iff (P\wedge Q)\vee (P\wedge R)\end{array}$$

(77)

$$\begin{array}{rl}P\nrightarrow (Q\uparrow R)& \iff P\wedge \mathrm{\neg }(Q\uparrow R)\\ & \iff P\wedge (Q\wedge R)\\ & \iff (P\wedge Q)\wedge R\end{array}$$

(78)

$$\begin{array}{rl}P\nrightarrow (Q\nrightarrow R)& \iff P\wedge \mathrm{\neg }(Q\wedge \mathrm{\neg }R)\\ & \iff P\wedge (\mathrm{\neg }Q\vee R)\\ & \iff (P\wedge \mathrm{\neg }Q)\vee (P\wedge R)\\ & \iff (P\nrightarrow Q)\vee (P\wedge R)\end{array}$$

(79)

$$\begin{array}{rl}P\nrightarrow (Q\nleftarrow R)& \iff P\wedge \mathrm{\neg }(\mathrm{\neg }Q\wedge R)\\ & \iff P\wedge (Q\vee \mathrm{\neg }R)\\ & \iff (P\wedge Q)\vee (P\wedge \mathrm{\neg }R)\\ & \iff (P\wedge Q)\vee (P\nrightarrow R)\end{array}$$

(80)

$$\begin{array}{rl}P\nrightarrow (Q\nleftrightarrow R)& \iff P\wedge \mathrm{\neg }(Q\nleftrightarrow R)\\ & \iff \mathrm{\neg }\{\mathrm{\neg }P\vee (\mathrm{\neg }Q\leftrightarrow R)\}\\ & \iff \mathrm{\neg }\{(\mathrm{\neg }P\vee \mathrm{\neg }Q)\leftrightarrow (\mathrm{\neg }P\vee R)\}\\ & \iff (P\uparrow Q)\nleftrightarrow (P\to R)\end{array}$$ $$\begin{array}{rl}P\nrightarrow (Q\nleftrightarrow R)& \iff P\nrightarrow (\mathrm{\neg }Q\nleftrightarrow \mathrm{\neg }R)\\ & \iff (P\uparrow \mathrm{\neg }Q)\nleftrightarrow (P\to \mathrm{\neg }R)\\ & \iff (P\to Q)\nleftrightarrow (P\uparrow R)\\ & \iff (P\nrightarrow Q)\nleftrightarrow (P\wedge R)\end{array}$$

(81)

$$\begin{array}{rl}P\nleftarrow (Q\vee R)& \iff \mathrm{\neg }P\wedge (Q\vee R)\\ & \iff (\mathrm{\neg }P\wedge Q)\vee (\mathrm{\neg }P\wedge R)\\ & \iff (P\nleftarrow Q)\vee (P\nleftarrow R)\end{array}$$

(82)

$$\begin{array}{rl}P\nleftarrow (Q\wedge R)& \iff \mathrm{\neg }P\wedge (Q\wedge R)\\ & \iff (\mathrm{\neg }P\wedge Q)\wedge R\\ & \iff (P\nleftarrow Q)\wedge R\end{array}$$

(83)

$$\begin{array}{rl}P\nleftarrow (Q\to R)& \iff \mathrm{\neg }P\wedge (\mathrm{\neg }Q\vee R)\\ & \iff (\mathrm{\neg }P\wedge \mathrm{\neg }Q)\vee (\mathrm{\neg }P\wedge R)\\ & \iff (P\downarrow Q)\vee (P\nleftarrow R)\end{array}$$

(84)

$$\begin{array}{rl}P\nleftarrow (Q\leftarrow R)& \iff \mathrm{\neg }P\wedge (Q\vee \mathrm{\neg }R)\\ & \iff (\mathrm{\neg }P\wedge Q)\vee (\mathrm{\neg }P\wedge \mathrm{\neg }R)\\ & \iff (P\nleftarrow Q)\vee (P\downarrow R)\end{array}$$

(85)

(5)より、
$$\begin{array}{rl}P\nleftarrow (Q\leftrightarrow R)& \iff \mathrm{\neg }P\wedge (Q\leftrightarrow R)\\ & \iff P\downarrow (Q\nleftrightarrow R)\\ & \iff \mathrm{\neg }\{P\vee (\mathrm{\neg }Q\leftrightarrow R)\}\\ & \iff \mathrm{\neg }\{(P\vee \mathrm{\neg }Q)\leftrightarrow (P\vee R)\}\\ & \iff (P\leftarrow Q)\nleftrightarrow (P\vee R)\\ & \iff (P\nleftarrow Q)\leftrightarrow (P\vee R)\end{array}$$ $$\begin{array}{rl}P\nleftarrow (Q\leftrightarrow R)& \iff P\nleftarrow (\mathrm{\neg }Q\leftrightarrow \mathrm{\neg }R)\\ & \iff (P\nleftarrow \mathrm{\neg }Q)\leftrightarrow (P\vee \mathrm{\neg }R)\\ & \iff (P\downarrow Q)\leftrightarrow (P\leftarrow R)\\ & \iff (P\vee Q)\nleftrightarrow (P\leftarrow R)\end{array}$$

(86)

$$\begin{array}{rl}P\nleftarrow (Q\downarrow R)& \iff \mathrm{\neg }P\wedge (Q\downarrow R)\\ & \iff \mathrm{\neg }P\wedge (\mathrm{\neg }Q\wedge \mathrm{\neg }R)\\ & \iff (\mathrm{\neg }P\wedge \mathrm{\neg }Q)\wedge \mathrm{\neg }R\\ & \iff (P\downarrow Q)\nrightarrow R\end{array}$$

(87)

$$\begin{array}{rl}P\nleftarrow (Q\uparrow R)& \iff \mathrm{\neg }P\wedge (Q\uparrow R)\\ & \iff \mathrm{\neg }P\wedge (\mathrm{\neg }Q\vee \mathrm{\neg }R)\\ & \iff (\mathrm{\neg }P\wedge \mathrm{\neg }Q)\vee (\mathrm{\neg }P\wedge \mathrm{\neg }R)\\ & \iff (P\downarrow Q)\vee (P\downarrow R)\end{array}$$

(88)

$$\begin{array}{rl}P\nleftarrow (Q\nrightarrow R)& \iff \mathrm{\neg }P\wedge (Q\nrightarrow R)\\ & \iff \mathrm{\neg }P\wedge (Q\wedge \mathrm{\neg }R)\\ & \iff (\mathrm{\neg }P\wedge Q)\wedge \mathrm{\neg }R\\ & \iff (P\nleftarrow Q)\nrightarrow R\end{array}$$

(89)

$$\begin{array}{rl}P\nleftarrow (Q\nleftarrow R)& \iff \mathrm{\neg }P\wedge (Q\nleftarrow R)\\ & \iff \mathrm{\neg }P\wedge (\mathrm{\neg }Q\wedge R)\\ & \iff (\mathrm{\neg }P\wedge \mathrm{\neg }Q)\wedge R\\ & \iff (P\downarrow Q)\wedge R\end{array}$$

(90)

(5)より、
$$\begin{array}{rl}P\nleftarrow (Q\nleftrightarrow R)& \iff \mathrm{\neg }P\wedge (Q\nleftrightarrow R)\\ & \iff \mathrm{\neg }\{P\vee (Q\leftrightarrow R)\}\\ & \iff \mathrm{\neg }\{(P\vee Q)\leftrightarrow (P\vee R)\}\\ & \iff (P\vee Q)\nleftrightarrow (P\vee R)\end{array}$$ $$\begin{array}{rl}P\nleftarrow (Q\nleftrightarrow R)& \iff P\nleftarrow (\mathrm{\neg }Q\nleftrightarrow \mathrm{\neg }R)\\ & \iff (P\vee \mathrm{\neg }Q)\nleftrightarrow (P\vee \mathrm{\neg }R)\\ & \iff (P\leftarrow Q)\nleftrightarrow (P\leftarrow R)\end{array}$$

(91)

(41)より、
$$\begin{array}{rl}P\nleftrightarrow (Q\vee R)& \iff \mathrm{\neg }P\leftrightarrow (Q\vee R)\\ & \iff \{(\mathrm{\neg }P\leftrightarrow Q)\vee (\mathrm{\neg }P\wedge R)\}\wedge (\mathrm{\neg }P\leftarrow R)\\ & \iff \{(P\nleftrightarrow Q)\vee (P\nleftarrow R)\}\wedge (P\uparrow R)\end{array}$$ $$\begin{array}{rl}P\nleftrightarrow (Q\vee R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\nleftrightarrow (Q\vee R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\nleftrightarrow Q)\vee (\mathrm{\neg }P\nleftarrow R)\}\wedge (\mathrm{\neg }P\uparrow R)\}\\ & \iff \mathrm{\neg }\{\{(P\leftrightarrow Q)\vee (P\wedge R)\}\wedge (P\leftarrow R)\}\\ & \iff \{(P\nleftrightarrow Q)\wedge (P\uparrow R)\}\vee (P\nleftarrow R)\end{array}$$

(92)

(41)より、
$$\begin{array}{rl}P\nleftrightarrow (Q\wedge R)& \iff P\leftrightarrow (\mathrm{\neg }Q\vee \mathrm{\neg }R)\\ & \iff \{(P\leftrightarrow \mathrm{\neg }Q)\vee (P\wedge \mathrm{\neg }R)\}\wedge (P\leftarrow \mathrm{\neg }R)\\ & \iff \{(P\nleftrightarrow Q)\vee (P\nrightarrow R)\}\wedge (P\vee R)\end{array}$$ $$\begin{array}{rl}P\nleftrightarrow (Q\wedge R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\nleftrightarrow (Q\wedge R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\nleftrightarrow Q)\vee (\mathrm{\neg }P\nrightarrow R)\}\wedge (\mathrm{\neg }P\vee R)\}\\ & \iff \mathrm{\neg }\{\{(P\leftrightarrow Q)\vee (P\downarrow R)\}\wedge (P\to R)\}\\ & \iff \{(P\nleftrightarrow Q)\wedge (P\vee R)\}\vee (P\nrightarrow R)\end{array}$$

(93)

(41)より、
$$\begin{array}{rl}P\nleftrightarrow (Q\to R)& \iff \mathrm{\neg }P\leftrightarrow (\mathrm{\neg }Q\vee R)\\ & \iff \{(\mathrm{\neg }P\leftrightarrow \mathrm{\neg }Q)\vee (\mathrm{\neg }P\wedge R)\}\wedge (\mathrm{\neg }P\leftarrow R)\\ & \iff \{(P\leftrightarrow Q)\vee (P\nleftarrow R)\}\wedge (P\uparrow R)\\ & \iff \{(P\nleftrightarrow Q)\uparrow (P\leftarrow R)\}\wedge (P\uparrow R)\end{array}$$ $$\begin{array}{rl}P\nleftrightarrow (Q\to R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\nleftrightarrow (Q\to R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\nleftrightarrow Q)\uparrow (\mathrm{\neg }P\leftarrow R)\}\wedge (\mathrm{\neg }P\uparrow R)\}\\ & \iff \mathrm{\neg }\{\{(P\leftrightarrow Q)\uparrow (P\uparrow R)\}\wedge (P\leftarrow R)\}\\ & \iff \{(P\leftrightarrow Q)\wedge (P\uparrow R)\}\vee (P\nleftarrow R)\\ & \iff \{(P\nleftrightarrow Q)\downarrow (P\wedge R)\}\vee (P\nleftarrow R)\end{array}$$

(94)

(41)より、
$$\begin{array}{rl}P\nleftrightarrow (Q\leftarrow R)& \iff \mathrm{\neg }P\leftrightarrow (Q\vee \mathrm{\neg }R)\\ & \iff \{(\mathrm{\neg }P\leftrightarrow Q)\vee (\mathrm{\neg }P\wedge \mathrm{\neg }R)\}\wedge (\mathrm{\neg }P\leftarrow \mathrm{\neg }R)\\ & \iff \{(P\nleftrightarrow Q)\vee (P\downarrow R)\}\wedge (P\to R)\end{array}$$ $$\begin{array}{rl}P\nleftrightarrow (Q\leftarrow R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\nleftrightarrow (Q\leftarrow R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\nleftrightarrow Q)\vee (\mathrm{\neg }P\downarrow R)\}\wedge (\mathrm{\neg }P\to R)\}\\ & \iff \mathrm{\neg }\{\{(P\leftrightarrow Q)\vee (P\nrightarrow R)\}\wedge (P\vee R)\}\\ & \iff \{(P\nleftrightarrow Q)\wedge (P\to R)\}\vee (P\downarrow R)\end{array}$$

(95)

(45)より、
$$\begin{array}{rl}P\nleftrightarrow (Q\leftrightarrow R)& \iff \mathrm{\neg }P\leftrightarrow (Q\leftrightarrow R)\\ & \iff (\mathrm{\neg }P\leftrightarrow Q)\leftrightarrow R\\ & \iff (P\nleftrightarrow Q)\leftrightarrow R\end{array}$$

(96)

(41)より、
$$\begin{array}{rl}P\nleftrightarrow (Q\downarrow R)& \iff P\leftrightarrow (Q\vee R)\\ & \iff \{(P\leftrightarrow Q)\vee (P\wedge R)\}\wedge (P\leftarrow R)\\ & \iff \{(P\nleftrightarrow Q)\uparrow (P\uparrow R)\}\wedge (P\leftarrow R)\end{array}$$ $$\begin{array}{rl}P\nleftrightarrow (Q\downarrow R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\nleftrightarrow (Q\downarrow R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\nleftrightarrow Q)\uparrow (\mathrm{\neg }P\uparrow R)\}\wedge (\mathrm{\neg }P\leftarrow R)\}\\ & \iff \mathrm{\neg }\{\{(P\leftrightarrow Q)\uparrow (P\leftarrow R)\}\wedge (P\uparrow R)\}\\ & \iff \{(P\leftrightarrow Q)\wedge (P\leftarrow R)\}\vee (P\wedge R)\\ & \iff \{(P\nleftrightarrow Q)\downarrow (P\nleftarrow R)\}\vee (P\wedge R)\end{array}$$

(97)

(41)より、
$$\begin{array}{rl}P\nleftrightarrow (Q\uparrow R)& \iff \mathrm{\neg }P\leftrightarrow (\mathrm{\neg }Q\vee \mathrm{\neg }R)\\ & \iff \{(\mathrm{\neg }P\leftrightarrow \mathrm{\neg }Q)\vee (\mathrm{\neg }P\wedge \mathrm{\neg }R)\}\wedge (\mathrm{\neg }P\leftarrow \mathrm{\neg }R)\\ & \iff \{(P\leftrightarrow Q)\vee (P\downarrow R)\}\wedge (P\to R)\\ & \iff \{(P\nleftrightarrow Q)\uparrow (P\vee R)\}\wedge (P\to R)\end{array}$$ $$\begin{array}{rl}P\nleftrightarrow (Q\uparrow R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\nleftrightarrow (Q\uparrow R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\nleftrightarrow Q)\uparrow (\mathrm{\neg }P\vee R)\}\wedge (\mathrm{\neg }P\to R)\}\\ & \iff \mathrm{\neg }\{\{(P\leftrightarrow Q)\uparrow (P\to R)\}\wedge (P\vee R)\}\\ & \iff \{(P\leftrightarrow Q)\wedge (P\to R)\}\vee (P\downarrow R)\\ & \iff \{(P\nleftrightarrow Q)\downarrow (P\nrightarrow R)\}\vee (P\downarrow R)\end{array}$$

(98)

(41)より、
$$\begin{array}{rl}P\nleftrightarrow (Q\nrightarrow R)& \iff P\leftrightarrow (Q\to R)\\ & \iff P\leftrightarrow (\mathrm{\neg }Q\vee R)\\ & \iff \{(P\leftrightarrow \mathrm{\neg }Q)\wedge (P\leftarrow R)\}\vee (P\wedge R)\\ & \iff \{(P\nleftrightarrow Q)\wedge (P\leftarrow R)\}\vee (P\wedge R)\end{array}$$ $$\begin{array}{rl}P\nleftrightarrow (Q\nrightarrow R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\nleftrightarrow (Q\uparrow R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\nleftrightarrow Q)\wedge (\mathrm{\neg }P\leftarrow R)\}\vee (\mathrm{\neg }P\wedge R)\}\\ & \iff \mathrm{\neg }\{\{(P\leftrightarrow Q)\wedge (P\uparrow R)\}\vee (P\nleftarrow R)\}\\ & \iff \{(P\nleftrightarrow Q)\vee (P\wedge R)\}\wedge (P\leftarrow R)\end{array}$$

(99)

(41)より、
$$\begin{array}{rl}P\nleftrightarrow (Q\nleftarrow R)& \iff P\leftrightarrow (Q\leftarrow R)\\ & \iff P\leftrightarrow (Q\vee \mathrm{\neg }R)\\ & \iff \{(P\leftrightarrow Q)\vee (P\wedge \mathrm{\neg }R)\}\wedge (P\leftarrow \mathrm{\neg }R)\\ & \iff \{(P\leftrightarrow Q)\vee (P\nrightarrow R)\}\wedge (P\vee R)\\ & \iff \{(P\nleftrightarrow Q)\uparrow (P\to R)\}\wedge (P\vee R)\end{array}$$ $$\begin{array}{rl}P\nleftrightarrow (Q\nleftarrow R)& \iff \mathrm{\neg }\{\mathrm{\neg }P\nleftrightarrow (Q\nleftarrow R)\}\\ & \iff \mathrm{\neg }\{\{(\mathrm{\neg }P\nleftrightarrow Q)\uparrow (\mathrm{\neg }P\to R)\}\wedge (\mathrm{\neg }P\vee R)\}\\ & \iff \mathrm{\neg }\{\{(P\leftrightarrow Q)\uparrow (P\vee R)\}\wedge (P\to R)\}\\ & \iff \{(P\leftrightarrow Q)\wedge (P\vee R)\}\vee (P\nrightarrow R)\\ & \iff \{(P\nleftrightarrow Q)\downarrow (P\downarrow R)\}\vee (P\nrightarrow R)\end{array}$$

(100)

(45)より、
$$\begin{array}{rl}P\nleftrightarrow (Q\nleftrightarrow R)& \iff P\leftrightarrow \mathrm{\neg }(Q\nleftrightarrow R)\\ & \iff P\leftrightarrow (Q\leftrightarrow R)\\ & \iff (P\leftrightarrow Q)\leftrightarrow R\\ & \iff \mathrm{\neg }(P\leftrightarrow Q)\nleftrightarrow R\\ & \iff (P\nleftrightarrow Q)\nleftrightarrow R\end{array}$$