Denying the Antecedent/Corollary

Corollary to Denying the Antecedent
Let $p \implies q$ be a conditional statement.

Let its antecedent $p$ be false.

Then nothing can be inferred about the truth value of $q$.

Proof
By applying the Method of Truth Tables to the proposition, it is seen that $q$ can be either true or false, and both are consistent with both $\neg p$ and $p \implies q$.

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