Non-Equivalence as Disjunction of Negated Implications/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.

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