Rule of Explosion/Variant 1

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Theorem

$\vdash p \implies \paren {\neg p \implies q}$


Proof

By the tableau method of natural deduction:

$\vdash p \implies \paren {\neg p \implies q} $
Line Pool Formula Rule Depends upon Notes
1 1 $p$ Assumption (None)
2 2 $\neg p$ Assumption (None)
3 1, 2 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 1, 2
4 1, 2 $q$ Rule of Explosion: $\bot \mathcal E$ 3
5 1 $\neg p \implies q$ Rule of Implication: $\implies \mathcal I$ 2 – 4 Assumption 2 has been discharged
6 $p \implies \paren {\neg p \implies q}$ Rule of Implication: $\implies \mathcal I$ 1 – 5 Assumption 1 has been discharged

$\blacksquare$


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