# Rule of Explosion/Variant 2

## Theorem

- $\vdash \left({p \land \neg p}\right) \implies q$

## Proof

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $p \land \neg p$ | Assumption | (None) | ||

2 | 1 | $p$ | Rule of Simplification: $\land \mathcal E_1$ | 1 | ||

3 | 1 | $\neg p$ | Rule of Simplification: $\land \mathcal E_2$ | 1 | ||

4 | 1 | $p \lor q$ | Rule of Addition: $\lor \mathcal I_1$ | 2 | ||

5 | 1 | $q$ | Modus Tollendo Ponens $\mathrm{MTP}_{{{6}}}$ | 4, 3 |

$\blacksquare$