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Theorem cdeqel 3158
Description: Distribute conditional equality over elementhood. (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
cdeqel  |- CondEq ( x  =  y  ->  ( A  e.  C  <->  B  e.  D ) )

Proof of Theorem cdeqel
StepHypRef Expression
1 cdeqeq.1 . . . 4  |- CondEq ( x  =  y  ->  A  =  B )
21cdeqri 3148 . . 3  |-  ( x  =  y  ->  A  =  B )
3 cdeqeq.2 . . . 4  |- CondEq ( x  =  y  ->  C  =  D )
43cdeqri 3148 . . 3  |-  ( x  =  y  ->  C  =  D )
52, 4eleq12d 2505 . 2  |-  ( x  =  y  ->  ( A  e.  C  <->  B  e.  D ) )
65cdeqi 3147 1  |- CondEq ( x  =  y  ->  ( A  e.  C  <->  B  e.  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726  CondEqwcdeq 3145
This theorem is referenced by:  nfccdeq  3160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-cleq 2430  df-clel 2433  df-cdeq 3146
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