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Theorem cdeqel 2824
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 2814 . . 3  |-  ( x  =  y  ->  A  =  B )
3 cdeqeq.2 . . . 4  |- CondEq ( x  =  y  ->  C  =  D )
43cdeqri 2814 . . 3  |-  ( x  =  y  ->  C  =  D )
52, 4eleq12d 2155 . 2  |-  ( x  =  y  ->  ( A  e.  C  <->  B  e.  D ) )
65cdeqi 2813 1  |- CondEq ( x  =  y  ->  ( A  e.  C  <->  B  e.  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1287    e. wcel 1436  CondEqwcdeq 2811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-cleq 2078  df-clel 2081  df-cdeq 2812
This theorem is referenced by:  nfccdeq  2826
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