Rule of Implication/Sequent Form/Proof by Truth Table

Theorem

 * $\left({p \vdash q}\right) \vdash p \implies q$

Proof
We apply the Method of Truth Tables.

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

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