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Theorem pm2.65i 578
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  |-  ( ph  ->  ps )
pm2.65i.2  |-  ( ph  ->  -.  ps )
Assertion
Ref Expression
pm2.65i  |-  -.  ph

Proof of Theorem pm2.65i
StepHypRef Expression
1 pm2.65i.2 . . 3  |-  ( ph  ->  -.  ps )
2 pm2.65i.1 . . 3  |-  ( ph  ->  ps )
31, 2nsyl3 566 . 2  |-  ( ph  ->  -.  ph )
4 pm2.01 556 . 2  |-  ( (
ph  ->  -.  ph )  ->  -.  ph )
53, 4ax-mp 7 1  |-  -.  ph
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 554  ax-in2 555
This theorem is referenced by:  mt2  579  mto  598  pm5.19  632  noel  3255  0nelop  4012  elirr  4293  en2lp  4305  soirri  4746  0neqopab  5577  fzp1disj  9043  fzonel  9117  fzouzdisj  9137
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