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Theorem pm2.65i 568
Description: Inference rule for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
Hypotheses
Ref Expression
pm2.65i.1 (𝜑𝜓)
pm2.65i.2 (𝜑 → ¬ 𝜓)
Assertion
Ref Expression
pm2.65i ¬ 𝜑

Proof of Theorem pm2.65i
StepHypRef Expression
1 pm2.65i.2 . . 3 (𝜑 → ¬ 𝜓)
2 pm2.65i.1 . . 3 (𝜑𝜓)
31, 2nsyl3 556 . 2 (𝜑 → ¬ 𝜑)
4 pm2.01 546 . 2 ((𝜑 → ¬ 𝜑) → ¬ 𝜑)
53, 4ax-mp 7 1 ¬ 𝜑
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-in1 544  ax-in2 545
This theorem is referenced by:  mt2  569  mto  588  pm5.19  622  noel  3225  0nelop  3982  elirr  4260  en2lp  4272  soirri  4706  0neqopab  5537  fzp1disj  8909  fzonel  8983  fzouzdisj  9003
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