| $(M \leftrightarrow N) \rightarrow O$ |
| $\sim A\ v\ (B\ \&\ D)$ |
| $B \rightarrow (O \rightarrow P)$ |
| $\sim \sim A$ |
| $\therefore (M \leftrightarrow N) \rightarrow P$ |
| $(1)\ (M \leftrightarrow N) \rightarrow O$ | $P$ |
| $(2)\ \sim\ A\ v\ (B\ \&\ D)\ \sim $ | $P$ |
| $(3)\ B\ \rightarrow (O\ \rightarrow\ P)$ | $P$ |
| $(4)\ \sim\ A$ | $P/ \therefore\ \sim (M \leftrightarrow N) \rightarrow P$ |
| $(5)\ B\ \&\ D$ | $2, 4, DS$ |
| $(6) \ B$ | $5,$ Simp. |
| $(7)\ 0\ \rightarrow\ P$ | $3, 6, MP$ |
| $(8)\ (M\ \therefore\ N) \rightarrow\ P$ | $1, 7, HS$ |
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| $(H\ \&\ K)\ \rightarrow\ (J\ v\ K)$ |
| $\sim\ E\ \&\ \sim\ F$ |
| $F\ v\ \sim\ (J\ v\ K)$ |
| $\sim\ (H\ \&\ K)\ \rightarrow\ H$ |
| $H\ \&\ \sim\ E$ |
| $J\ v\ (J\ \&\ K)$ |
| $J \rightarrow D$ |
| $\sim\ D$ |
| $\therefore K\ v\ M$ |
| (G $\rightarrow$ H) $\rightarrow$ (I $\leftrightarrow$ J) |
| K v ~(L $\rightarrow$ M) |
| (G $\rightarrow$ H) v ~ K |
| N $\rightarrow$ (L $\rightarrow$ M) |
| ~ (I J) |
| $\therefore$~ N & ~ K |
| $(P\ v\ Q)\ \rightarrow\ (T\ v\ S)$ |
| $\sim\ N\ \& \sim\ M$ |
| $N\ v\ \sim\ (T\ v\ S)$ |
| $H\ \rightarrow\ (P\ v\ Q)$ |
| $\therefore \sim\ H\ \&\ \sim\ M$ |