Conjunction with Negative Equivalent to Negation of Implication/Formulation 1/Proof by Truth Table

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.

$\begin{array}{|cccc||cccc|} \hline p & \land & \neg & q & \neg & (p & \implies & q) \\ \hline \F & \F & \T & \F & \F & \F & \T & \F \\ \F & \F & \F & \T & \F & \F & \T & \T \\ \T & \T & \T & \F & \T & \T & \F & \F \\ \T & \F & \F & \T & \F & \T & \T & \T \\ \hline \end{array}$