Theorem pm2.65i 116

Description: Inference rule for proof by contradiction.

Hypotheses

Ref	Expression
pm2.65i.1	⊢ (φ → ψ)
pm2.65i.2	⊢ (φ → ¬ ψ)

Assertion

Ref	Expression
pm2.65i	⊢ ¬ φ

Proof of Theorem pm2.65i

Step	Hyp	Ref	Expression
1		pm2.65i.2	. . 3 ⊢ (φ → ¬ ψ)
2		pm2.65i.1	. . 3 ⊢ (φ → ψ)
3	1, 2	nsyl 102	. 2 ⊢ (φ → ¬ φ)
4		pm2.01 80	. 2 ⊢ ((φ → ¬ φ) → ¬ φ)
5	3, 4	ax-mp 6	1 ⊢ ¬ φ

Syntax hints:  ¬ wn 1   → wi 2

This theorem is referenced by:  canth 2945  cardprc 3667

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

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