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  3450  0nelop  4277  elirr  4573  en2lp  4586  soirri  5060  canth  5871  0neqopab  5963  fzp1disj  10146  fzonel  10227  fzouzdisj  10247  4sqlem17  12545  lgsval2lem  15126  bj-imnimnn  15230  nnnotnotr  15482