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Theorem imimorbdc 829
Description: Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.)
Assertion
Ref Expression
imimorbdc  |-  (DECID  ps  ->  ( ( ( ps  ->  ch )  ->  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  \/  ch ) ) ) )

Proof of Theorem imimorbdc
StepHypRef Expression
1 dfor2dc 828 . . 3  |-  (DECID  ps  ->  ( ( ps  \/  ch ) 
<->  ( ( ps  ->  ch )  ->  ch )
) )
21imbi2d 228 . 2  |-  (DECID  ps  ->  ( ( ph  ->  ( ps  \/  ch ) )  <-> 
( ph  ->  ( ( ps  ->  ch )  ->  ch ) ) ) )
3 bi2.04 246 . 2  |-  ( ( ( ps  ->  ch )  ->  ( ph  ->  ch ) )  <->  ( ph  ->  ( ( ps  ->  ch )  ->  ch )
) )
42, 3syl6rbbr 197 1  |-  (DECID  ps  ->  ( ( ( ps  ->  ch )  ->  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  \/  ch ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    \/ 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-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by: (None)
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