Theorem pm2.65i 644

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 631	. 2 ⊢ (𝜑 → ¬ 𝜑)
4		pm2.01 621	. 2 ⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑)
5	3, 4	ax-mp 5	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 619  ax-in2 620

This theorem is referenced by:  mt2  645  mto  668  pm5.19  714  noel  3516  0nelop  4369  elirr  4668  en2lp  4681  soirri  5162  canth  6009  0neqopab  6106  fczsupp0  6472  fzp1disj  10439  fzonel  10520  fzouzdisj  10541  hashfibclem  11234  4sqlem17  13133  lgsval2lem  16012  bj-imnimnn  16649  nnnotnotr  16899