# Rule of Explosion/Variant 2/Proof 1

## Theorem

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

## Proof

By the tableau method of natural deduction:

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

$\blacksquare$