Theorem pm2.65i 640

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 627	. 2 ⊢ (𝜑 → ¬ 𝜑)
4		pm2.01 617	. 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 615  ax-in2 616

This theorem is referenced by:  mt2  641  mto  663  pm5.19  707  noel  3451  0nelop  4278  elirr  4574  en2lp  4587  soirri  5061  canth  5872  0neqopab  5964  fzp1disj  10149  fzonel  10230  fzouzdisj  10250  4sqlem17  12548  lgsval2lem  15167  bj-imnimnn  15300  nnnotnotr  15552