Conjunction iff Biconditional of Biconditional with Disjunction

Theorem

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

Proof
We apply the Method of Truth Tables.

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

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