Rule of Idempotence

Context
Natural deduction.

Definition
This rule is two-fold:

$$p \dashv \vdash p \land p$$
 * Conjunction is idempotent:

$$p \dashv \vdash p \lor p$$
 * Disjunction is idempotent:

Its abbreviation in a tableau proof is $$\textrm{Idemp}$$.

Proof
These are proved by the tableau method:

$$p \vdash p \land p$$:

$$p \land p \vdash p$$:

$$p \vdash p \lor p$$

$$p \lor p \vdash p$$