Theorem pm2.65d 166

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 132	. 2 ⊢ (φ → (ψ → ¬ ψ))
4	3	pm2.01d 161	1 ⊢ (φ → ¬ ψ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  mtod  168  pm2.65da  559