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Theorem pm2.65i 629
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 (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 616 . 2 (𝜑 → ¬ 𝜑)
4 pm2.01 606 . 2 ((𝜑 → ¬ 𝜑) → ¬ 𝜑)
53, 4ax-mp 5 1 ¬ 𝜑
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 604  ax-in2 605
This theorem is referenced by:  mt2  630  mto  652  pm5.19  696  noel  3411  0nelop  4223  elirr  4515  en2lp  4528  soirri  4995  canth  5793  0neqopab  5881  fzp1disj  10009  fzonel  10089  fzouzdisj  10109  bj-imnimnn  13513  nnnotnotr  13765
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