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Theorem cdeqim 3198
Description: Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypotheses
Ref Expression
cdeqnot.1  |- CondEq ( x  =  y  ->  ( ph 
<->  ps ) )
cdeqim.1  |- CondEq ( x  =  y  ->  ( ch 
<->  th ) )
Assertion
Ref Expression
cdeqim  |- CondEq ( x  =  y  ->  (
( ph  ->  ch )  <->  ( ps  ->  th )
) )

Proof of Theorem cdeqim
StepHypRef Expression
1 cdeqnot.1 . . . 4  |- CondEq ( x  =  y  ->  ( ph 
<->  ps ) )
21cdeqri 3191 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3 cdeqim.1 . . . 4  |- CondEq ( x  =  y  ->  ( ch 
<->  th ) )
43cdeqri 3191 . . 3  |-  ( x  =  y  ->  ( ch 
<->  th ) )
52, 4imbi12d 320 . 2  |-  ( x  =  y  ->  (
( ph  ->  ch )  <->  ( ps  ->  th )
) )
65cdeqi 3190 1  |- CondEq ( x  =  y  ->  (
( ph  ->  ch )  <->  ( ps  ->  th )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184  CondEqwcdeq 3188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-cdeq 3189
This theorem is referenced by: (None)
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