Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication

Theorem

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

Proof

By the tableau method of natural deduction:

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

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p \lor p$	Premise	(None)
2		2	$p$	Assumption	(None)
3		1	$p$	Proof by Cases: $\text{PBC}$	1, 2 – 2, 2 – 2	Assumptions 2 and 2 have been discharged
4			$\left({p \lor p}\right) \implies p$	Rule of Implication: $\implies \II$	1 – 3	Assumption 1 has been discharged

$\blacksquare$

Also see

• This is an axiom of the following proof system:
  • Definition:Hilbert Proof System/Instance 2