Conditional iff Biconditional of Antecedent with Conjunction

Theorem

 * $p \implies q \dashv \vdash p \iff \paren {p \land q}$

Proof
We apply the Method of Truth Tables.

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

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