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Theorem pm2.65i 567
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  568  mto  587  pm5.19  621  noel  3222  0nelop  3976  elirr  4224  en2lp  4232  soirri  4662  0neqopab  5492  fzp1disj  8712  fzonel  8786  fzouzdisj  8806
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