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Theorem pm2.61ddc 792
Description: Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.)
Hypotheses
Ref Expression
pm2.61ddc.1  |-  ( ph  ->  ( ps  ->  ch ) )
pm2.61ddc.2  |-  ( ph  ->  ( -.  ps  ->  ch ) )
Assertion
Ref Expression
pm2.61ddc  |-  (DECID  ps  ->  (
ph  ->  ch ) )

Proof of Theorem pm2.61ddc
StepHypRef Expression
1 df-dc 777 . 2  |-  (DECID  ps  <->  ( ps  \/  -.  ps ) )
2 pm2.61ddc.1 . . . 4  |-  ( ph  ->  ( ps  ->  ch ) )
32com12 30 . . 3  |-  ( ps 
->  ( ph  ->  ch ) )
4 pm2.61ddc.2 . . . 4  |-  ( ph  ->  ( -.  ps  ->  ch ) )
54com12 30 . . 3  |-  ( -. 
ps  ->  ( ph  ->  ch ) )
63, 5jaoi 669 . 2  |-  ( ( ps  \/  -.  ps )  ->  ( ph  ->  ch ) )
71, 6sylbi 119 1  |-  (DECID  ps  ->  (
ph  ->  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 662  DECID wdc 776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by:  bijadc  810
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