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  3496  0nelop  4338  elirr  4637  en2lp  4650  soirri  5129  canth  5964  0neqopab  6061  fzp1disj  10308  fzonel  10389  fzouzdisj  10410  4sqlem17  12973  lgsval2lem  15732  bj-imnimnn  16284  nnnotnotr  16535