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Theorem imimorbdc 882
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 bi2.04 247 . 2  |-  ( ( ( ps  ->  ch )  ->  ( ph  ->  ch ) )  <->  ( ph  ->  ( ( ps  ->  ch )  ->  ch )
) )
2 dfor2dc 881 . . 3  |-  (DECID  ps  ->  ( ( ps  \/  ch ) 
<->  ( ( ps  ->  ch )  ->  ch )
) )
32imbi2d 229 . 2  |-  (DECID  ps  ->  ( ( ph  ->  ( ps  \/  ch ) )  <-> 
( ph  ->  ( ( ps  ->  ch )  ->  ch ) ) ) )
41, 3bitr4id 198 1  |-  (DECID  ps  ->  ( ( ( ps  ->  ch )  ->  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  \/  ch ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 698  DECID wdc 820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699
This theorem depends on definitions:  df-bi 116  df-dc 821
This theorem is referenced by: (None)
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