# Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication

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## Theorem

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

## Proof

By the tableau method of natural deduction:

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 |

$\blacksquare$

## Also see

- This is an axiom of the following proof system: