Absorption Laws (Logic)/Conjunction Absorbs Disjunction/Forward Implication

Theorem

$p \land \left({p \lor q}\right) \vdash p$

Proof

By the tableau method of natural deduction:

$p \land \left({p \lor q}\right) \vdash p$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p \land \left({p \lor q}\right)$	Premise	(None)
2		1	$p$	Rule of Simplification: $\land \mathcal E_1$	1

$\blacksquare$

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): $\S 1.5$: Further Proofs: Résumé of Rules: Theorems $31 \ \text{(a)}$