Theorem pm2.65i 642

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 629	. 2 ⊢ (𝜑 → ¬ 𝜑)
4		pm2.01 619	. 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 617  ax-in2 618

This theorem is referenced by:  mt2  643  mto  666  pm5.19  711  noel  3495  0nelop  4335  elirr  4634  en2lp  4647  soirri  5126  canth  5961  0neqopab  6058  fzp1disj  10293  fzonel  10374  fzouzdisj  10395  4sqlem17  12951  lgsval2lem  15710  bj-imnimnn  16211  nnnotnotr  16462