Modus Ponendo Ponens/Sequent Form/Proof by Truth Table

Proof
We apply the Method of Truth Tables.

$\begin{array}{|c|ccc||c|} \hline p & p & \implies & q & q\\ \hline \F & \F & \T & \F & \F \\ \F & \F & \T & \T & \T \\ \T & \T & \F & \F & \F \\ \T & \T & \T & \T & \T \\ \hline \end{array}$

As can be seen, when $p$ is true, and so is $p \implies q$, then $q$ is also true.