| $A\ v\ B$ |
| $\sim\ A\ \&\ C$ |
| $\therefore (B\ v\ \sim\ D)\ \&\ C$ |
| $(1)\ A\ v\ B$ | $P$ |
| $(2)\ \sim\ A\ \&\ C$ | $P/\therefore (B\ v\ \sim\ D)\ \&\ C$ |
| $(3)\ \sim\ A$ | $2,$ Simp. |
| $(4)\ B$ | $1, 3, DS$ |
| $(5)\ B\ v\ \sim\ D$ | $4,$ Add. |
| $(6)\ C$ | $2,$ Simp. |
| $(7)\ (B\ v\ \sim\ D)\ \&\ C$ | $5, 6,$ Conj. |
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| $(P\ \&\ R)\ \&\ \sim T$ |
| $T\ v\ (F\ \&\ G)$ |
| $(F\ \&\ G )\ \rightarrow\ H$ |
| $\therefore (H\ \&\ P)\ v\ F$ |
| (P$\rightarrow$Q) & (R v S) |
| (R v S) $\rightarrow$ ~ L |
| L v (M & N) |
| $\therefore$ [(P $\rightarrow$ Q) & M] & ~ L |
| $P \rightarrow Q$ |
| $R \rightarrow S$ |
| $( \sim\ Q\ \&\ A) \& (P\ v\ R)$ |
| $\therefore (S\ v\ Z) \& \sim\ Q$ |