Theorem pm2.65d 586

Description: Deduction rule for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.)

Hypotheses

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

Assertion

Ref	Expression
pm2.65d	⊢ (𝜑 → ¬ 𝜓)

Proof of Theorem pm2.65d

Step	Hyp	Ref	Expression
1		pm2.65d.2	. . 3 ⊢ (𝜑 → (𝜓 → ¬ 𝜒))
2		pm2.65d.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	nsyld 577	. 2 ⊢ (𝜑 → (𝜓 → ¬ 𝜓))
4	3	pm2.01d 548	1 ⊢ (𝜑 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-in1 544  ax-in2 545

This theorem is referenced by:  pm2.65da  587  mtod  589