Rule of Idempotence/Conjunction

Definition
The conjunction operator is idempotent:

Conjunction

 * $p \dashv \vdash p \land p$

Conjunction

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

Proof
By the tableau method of natural deduction:

Proof by Truth Table
We apply the Method of Truth Tables to the propositions.

As can be seen by inspection, the truth values under the main connectives match for each model.

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