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Theorem bijadc 776
Description: Combine antecedents into a single biconditional. This inference is reminiscent of jadc 760. (Contributed by Jim Kingdon, 4-May-2018.)
Hypotheses
Ref Expression
bijadc.1  |-  ( ph  ->  ( ps  ->  ch ) )
bijadc.2  |-  ( -. 
ph  ->  ( -.  ps  ->  ch ) )
Assertion
Ref Expression
bijadc  |-  (DECID  ps  ->  ( ( ph  <->  ps )  ->  ch ) )

Proof of Theorem bijadc
StepHypRef Expression
1 bi2 121 . . 3  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
2 bijadc.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
31, 2syli 33 . 2  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ch ) )
4 bi1 111 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
54con3d 561 . . 3  |-  ( (
ph 
<->  ps )  ->  ( -.  ps  ->  -.  ph )
)
6 bijadc.2 . . 3  |-  ( -. 
ph  ->  ( -.  ps  ->  ch ) )
75, 6syli 33 . 2  |-  ( (
ph 
<->  ps )  ->  ( -.  ps  ->  ch )
)
83, 7pm2.61ddc 758 1  |-  (DECID  ps  ->  ( ( ph  <->  ps )  ->  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 98  DECID wdc 742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630
This theorem depends on definitions:  df-bi 110  df-dc 743
This theorem is referenced by: (None)
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