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: $\implies \mathcal I$ 1 – 3 Assumption 1 has been discharged

$\blacksquare$