# Rule of Simplification/Sequent Form/Formulation 2/Proof 2

## Theorem

$(1): \quad \vdash p \land q \implies p$
$(2): \quad \vdash p \land q \implies q$

## Proof

We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connective are $T$ for all boolean interpretations.

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

$\blacksquare$