Conditional iff Biconditional of Consequent with Disjunction

Theorem

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

Proof
We apply the Method of Truth Tables.

As can be seen by inspection, the appropriate truth values match for all boolean interpretations.

$\begin{array}{|ccc||ccccc|} \hline p & \implies & q & q & \iff & (p & \lor & q) \\ \hline F & T & F & F & T & F & F & F \\ F & T & T & T & T & F & T & T \\ T & F & F & F & F & T & T & F \\ T & T & T & T & T & T & T & T \\ \hline \end{array}$