Rule of Idempotence/Disjunction/Formulation 2/Forward Implication

Theorem

$\vdash p \implies \left({p \lor p}\right)$

Proof

By the tableau method of natural deduction:

$p \implies \left({p \lor p}\right)$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p$	Assumption	(None)
2		1	$p \lor p$	Rule of Addition: $\lor \II_1$	1
3			$p \implies \left({p \lor p}\right)$	Rule of Implication: $\implies \II$	1 – 2	Assumption 1 has been discharged

$\blacksquare$