| $P \rightarrow Q$ |
| $\sim Q\ v\ R$ |
| $\sim R$ |
| $\therefore (\sim P \& \sim R)\ v\ S$ |
| $(1)\ P \rightarrow Q$ | $P$ |
| $(2)\ \sim Q\ v\ R$ | $P$ |
| $(3)\ \sim R$ | $P/ \therefore (\sim P\ \&\ \sim R)\ v\ S$ |
| $(4)\ \sim Q$ | $2, 3, DS$ |
| $(5)\ \sim p$ | $1, 4, MT$ |
| $(6)\ \sim P\ \&\ \sim R$ | $5, 3,$ Conj. |
| $(7)\ (\sim P\ \&\ \sim R)\ v\ S$ | $6 $ Add. |
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| $A \rightarrow B$ |
| $C \rightarrow B$ |
| $(\sim\ A\ \&\ \sim \ C)\ \rightarrow\ (D\ \rightarrow\ E)$ |
| $(E\ \rightarrow\ G)\ \&\ (D\ v\ E)$ |
| $E\ v\ G$ |
| $(P \rightarrow Q)\ \&\ (R \rightarrow S)$ |
| $(Q \rightarrow T)\ \&\ (P\ v\ R)$ |
| $\sim T$ |
| $(T\ v\ S)\ \&\ \sim T$ |
| $R \rightarrow S$ |
| $P \rightarrow Q$ |
| $P\ v\ Q$ |
| $Q \rightarrow R$ |
| $\therefore (P \rightarrow R) \& (R\ v\ S)$ |