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Theorem cdeqeq 2999
Description: Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypotheses
Ref Expression
cdeqeq.1  |- CondEq ( x  =  y  ->  A  =  B )
cdeqeq.2  |- CondEq ( x  =  y  ->  C  =  D )
Assertion
Ref Expression
cdeqeq  |- CondEq ( x  =  y  ->  ( A  =  C  <->  B  =  D ) )

Proof of Theorem cdeqeq
StepHypRef Expression
1 cdeqeq.1 . . . 4  |- CondEq ( x  =  y  ->  A  =  B )
21cdeqri 2990 . . 3  |-  ( x  =  y  ->  A  =  B )
3 cdeqeq.2 . . . 4  |- CondEq ( x  =  y  ->  C  =  D )
43cdeqri 2990 . . 3  |-  ( x  =  y  ->  C  =  D )
52, 4eqeq12d 2310 . 2  |-  ( x  =  y  ->  ( A  =  C  <->  B  =  D ) )
65cdeqi 2989 1  |- CondEq ( x  =  y  ->  ( A  =  C  <->  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632  CondEqwcdeq 2987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-cleq 2289  df-cdeq 2988
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