The Boolean expression

$(\mathrm{p} \rightarrow \mathrm{q}) \wedge(\mathrm{q} \rightarrow \sim \mathrm{p})$

$\equiv(\sim \mathrm{p} \vee \mathrm{q}) \wedge(\sim \mathrm{q} \vee \sim \mathrm{p}) \quad\{\mathrm{p} \rightarrow \mathrm{q} \equiv \sim \mathrm{p} \vee \mathrm{q}\}$

$\equiv(\sim \mathrm{p} \vee \mathrm{q}) \wedge(\sim \mathrm{p} \vee \sim \mathrm{q}) \quad\{$ commutative property $\}$

$\equiv \sim \mathrm{p} \vee(\mathrm{q} \wedge \sim \mathrm{q}) \quad\{$ distributive property $\}$

$\equiv \sim \mathrm{p}$