Proof by Cases/Formulation 2/Forward Implication/Proof by Truth Table

Proof
We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, the truth values under the main connective are true for all boolean interpretations.

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