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  713  noel  3497  0nelop  4342  elirr  4641  en2lp  4654  soirri  5133  canth  5974  0neqopab  6071  fzp1disj  10320  fzonel  10401  fzouzdisj  10422  4sqlem17  13003  lgsval2lem  15768  bj-imnimnn  16395  nnnotnotr  16645