Theorem pm2.65i 165

Description: Inference rule for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)

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	1	con2i 112	. 2 ⊢ (ψ → ¬ φ)
3		pm2.65i.1	. . 3 ⊢ (φ → ψ)
4	3	con3i 127	. 2 ⊢ (¬ ψ → ¬ φ)
5	2, 4	pm2.61i 156	1 ⊢ ¬ φ

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:  mto  167  mt2  170  noel  3555