| $A \rightarrow B$ |
| $B \rightarrow S$ |
| $A\ \&\ T$ |
| $\therefore S\ \&\ T$ |
| $(1)\ A \rightarrow B$ | $P$ |
| $(2)\ B \rightarrow S$ | $P$ |
| $(3)\ A\ \&\ T$ | $P/ \therefore S\ \&\ T$ |
| $(4)\ A$ | $3,$ Simp. |
| $(5)\ B$ | $1, 4, MP$ |
| $(6)\ S$ | $2, 5, MP$ |
| $(7)\ T$ | $3,$ Simp. |
| $(8)\ S\ \&\ T$ | $6, 7,$ Conj. |
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| $(M \leftrightarrow N) \rightarrow O$ |
| $\sim A\ v\ (B\ \&\ D)$ |
| $B \rightarrow (O \rightarrow P)$ |
| $\sim \sim A$ |
| $\therefore (M \leftrightarrow N) \rightarrow P$ |
| $H \rightarrow I$ |
| $H\ \&\ J$ |
| $I \rightarrow G$ |
| $G\ \&\ J$ |