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Theorem cdeqim 2997
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 2990 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3 cdeqim.1 . . . 4  |- CondEq ( x  =  y  ->  ( ch 
<->  th ) )
43cdeqri 2990 . . 3  |-  ( x  =  y  ->  ( ch 
<->  th ) )
52, 4imbi12d 311 . 2  |-  ( x  =  y  ->  (
( ph  ->  ch )  <->  ( ps  ->  th )
) )
65cdeqi 2989 1  |- CondEq ( x  =  y  ->  (
( ph  ->  ch )  <->  ( ps  ->  th )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632  CondEqwcdeq 2987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-cdeq 2988
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