Conditional iff Biconditional of Antecedent with Conjunction

Theorem

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

Proof
We apply the Method of Truth Tables.

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

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