| $A \rightarrow B$ |
| $(A\ \&\ B) \rightarrow C$ |
| $A$ |
| $\therefore (A\ \&\ B)\ \&\ (A\ \&\ C)$ |
| $(1)\ A \rightarrow B$ | $P$ |
| $(2)\ (A\ \&\ B)\ \rightarrow C$ | $P$ |
| $(3)\ A$ | $P/\therefore (A\ \&\ B)\ \&\ (A\ \&\ C)$ |
| $(4)\ B$ | $1, 3, MP$ |
| $(5)\ A\ \&\ B$ | $3, 4,$ Conj. |
| $(6)\ C$ | $2, 5, MP$ |
| $(7)\ A\ \&\ C$ | $3, 6,$ Conj. |
| $(8)\ (A\ \&\ B)\ \&\ (A\ \& \ C)$ | $5, 7,$ Conj. |
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| $A \rightarrow B$ |
| $D\ v\ A$ |
| $E\ \rightarrow \sim\ D$ |
| $F\ v\ E$ |
| $\therefore\ A\ \&\ B$ |
| $R\rightarrow (S\ \&\ T)$ |
| $P\ v\ \sim (S\ \&\ T)$ |
| $Q\ \&\ \sim P$ |
| $\sim R\rightarrow (X\ \&\ Y)$ |
| $Q\ \&\ X$ |
| $M \rightarrow \sim (S \ \&\ T)$ |
| $\sim (A \ \&\ B)\ v\ \sim D$ |
| $\sim D (S \ \&\ T)$ |
| $M$ |
| $[\sim (A \ \&\ B) \ \&\ M)\ v\ D$ |