| $M \rightarrow N$ |
| $N \rightarrow O$ |
| $(M \rightarrow O) \rightarrow (N \rightarrow P)$ |
| $(N \rightarrow P) \rightarrow Q$ |
| $(Q\ v\ Z)\ \&\ Q$ |
| $(1)\ M \rightarrow N$ | $P$ |
| $(2)\ N \rightarrow 0$ | $P$ |
| $(3)\ (M \rightarrow 0) \rightarrow (N \rightarrow P)$ | $P$ |
| $(4)\ (N \rightarrow P) \rightarrow Q$ | $P/ (Q\ v\ Z)\ \&\ Q$ |
| $(5)\ (M \rightarrow 0) \rightarrow Q$ | $3, 4, HS$ |
| $(6)\ M \rightarrow 0$ | $1, 2, HS$ |
| $(7)\ Q$ | $5, 6, MP$ |
| $(8)\ Q\ v\ Z$ | $7,$ Add. |
| $(9)\ (Q\ v\ Z)\ \&\ Q$ | $8, 7,$ Conj. |
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| $A \rightarrow B$ |
| $(A\ \&\ B) \rightarrow C$ |
| $A$ |
| $\therefore (A\ \&\ B)\ \&\ (A\ \&\ C)$ |
| $J \rightarrow K$ |
| $J\ v\ (K\ v\ \sim\ L)$ |
| $\sim K$ |
| $\therefore \sim L\ \&\ \sim K$ |
| $R\ \rightarrow\ (A\ \&\ B)$ |
| $P\ v\ \sim\ (S\ \&\ T)$ |
| $\sim\ T\ \&\ \sim\ P$ |
| $(A\ \&\ B)\ \rightarrow\ (S\ \&\ T)$ |
| $(\sim\ R\ \&\ \sim\ T)\ v\ D$ |
| $(M \leftrightarrow N) \rightarrow O$ |
| $\sim A\ v\ (B\ \&\ D)$ |
| $B \rightarrow (O \rightarrow P)$ |
| $\sim \sim A$ |
| $\therefore (M \leftrightarrow N) \rightarrow P$ |