The logical statement
\([\sim(\sim \mathrm{p} \vee \mathrm{q}) \vee(\mathrm{p} \wedge \mathrm{r})] \wedge(\sim \mathrm{q} \wedge \mathrm{r})\) is equivalent to

\(\begin{aligned} & {[\sim(\sim p \vee q) \vee(p \wedge r)] \wedge(\sim q \wedge r)} \\ & \equiv[(p \wedge \sim q) \vee(p \wedge r)] \wedge(\sim q \wedge r) \\ & \text {...[De Morgan's law] } \\ & \equiv \mathrm{p} \wedge(\sim q \vee r) \wedge(\sim q \wedge r) \ldots \text {. [Distributive law] } \\ & \equiv \mathrm{p} \wedge[(\sim \mathrm{q} \vee \mathrm{r}) \wedge \sim \mathrm{q}] \wedge \mathrm{r} \ldots \text {... Associative law] } \\ & \equiv p \wedge(\sim q) \wedge r \quad \text {... [Absorption law] } \\ & \equiv(p \wedge r) \wedge \sim q \quad \ldots[\text { Commutative law }] \\ & \end{aligned}\)