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Theorem cdeqeq 2932
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 2923 . . 3  |-  ( x  =  y  ->  A  =  B )
3 cdeqeq.2 . . . 4  |- CondEq ( x  =  y  ->  C  =  D )
43cdeqri 2923 . . 3  |-  ( x  =  y  ->  C  =  D )
52, 4eqeq12d 2172 . 2  |-  ( x  =  y  ->  ( A  =  C  <->  B  =  D ) )
65cdeqi 2922 1  |- CondEq ( x  =  y  ->  ( A  =  C  <->  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1335  CondEqwcdeq 2920
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-5 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-cleq 2150  df-cdeq 2921
This theorem is referenced by: (None)
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