Theorem pm2.65i 648

Description: Inference 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		pm2.65i.1	. . 3 ⊢ (𝜑 → 𝜓)
3	1, 2	nsyl3 635	. 2 ⊢ (𝜑 → ¬ 𝜑)
4		pm2.01 625	. 2 ⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑)
5	3, 4	ax-mp 5	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-in1 623  ax-in2 624

This theorem is used by:  mt2  649  mto  672  pm5.19  718  noel  3525  0nelop  4388  elirr  4688  en2lp  4701  soirri  5182  canth  6036  0neqopab  6133  fczsupp0  6499  fzp1disj  10498  fzonel  10579  fzouzdisj  10600  hashfibclem  11298  4sqlem17  13209  lgsval2lem  16300  bj-imnimnn  16937  nnnotnotr  17187  als-no-surprise  17319