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Theorem jadc 771
Description: Inference forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 25-Mar-2018.)
Hypotheses
Ref Expression
jadc.1  |-  (DECID  ph  ->  ( -.  ph  ->  ch )
)
jadc.2  |-  ( ps 
->  ch )
Assertion
Ref Expression
jadc  |-  (DECID  ph  ->  ( ( ph  ->  ps )  ->  ch ) )

Proof of Theorem jadc
StepHypRef Expression
1 jadc.2 . . 3  |-  ( ps 
->  ch )
21imim2i 12 . 2  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ch )
)
3 jadc.1 . . 3  |-  (DECID  ph  ->  ( -.  ph  ->  ch )
)
4 pm2.6dc 770 . . 3  |-  (DECID  ph  ->  ( ( -.  ph  ->  ch )  ->  ( ( ph  ->  ch )  ->  ch ) ) )
53, 4mpd 13 . 2  |-  (DECID  ph  ->  ( ( ph  ->  ch )  ->  ch ) )
62, 5syl5 32 1  |-  (DECID  ph  ->  ( ( ph  ->  ps )  ->  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  pm5.71dc  879
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