Proof by Cases/Formulation 1/Proof

Proof
We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.

$\begin{array}{|ccccccc||ccccc|} \hline (p & \implies & r) & \land & (q & \implies & r) & (p & \lor & q) & \implies & r \\ \hline \F & \T & \F & \T & \F & \T & \F & \F & \F & \F & \T & \F \\ \F & \T & \T & \T & \F & \T & \T & \F & \F & \F & \T & \T \\ \F & \T & \F & \F & \T & \F & \F & \F & \T & \T & \F & \F \\ \F & \T & \T & \T & \T & \T & \T & \F & \T & \T & \T & \T \\ \T & \F & \F & \F & \F & \T & \F & \T & \T & \F & \F & \F \\ \T & \T & \T & \T & \F & \T & \T & \T & \T & \F & \T & \T \\ \T & \F & \F & \F & \T & \F & \F & \T & \T & \T & \F & \F \\ \T & \T & \T & \T & \T & \T & \T & \T & \T & \T & \T & \T \\ \hline \end{array}$