# Rule of Explosion/Variant 3

## Theorem

$p, \neg p \vdash q$

## Proof

This proof is derived in the context of the following proof system: Instance 2 of the Hilbert-style systems.

By the tableau method:

$p, \neg p \vdash q$
Line Pool Formula Rule Depends upon Notes
1 1 $p$ Assumption (None)
2 2 $\neg p$ Assumption (None)
3 $q \implies (p \lor q)$ Axiom $A2$
4 $\neg p \implies (q \lor \neg p)$ Rule $RST \, 1$ 3 $\neg p \, / \, q$, $q \, / \, p$
5 $q \lor \neg p$ Rule $RST \, 3$ 2, 4
6 $(q \lor \neg p) \implies (\neg p \lor q)$ Axiom $A3$, Rule $RST \, 1$ $\neg p \, / \, q$, $q \, / \, p$
7 $\neg p \lor q$ Rule $RST \, 3$ 5, 6
8 $p \implies q$ Rule $RST \, 2 \, (2)$
9 $q$ Rule $RST \, 3$ 1, 8

$\blacksquare$