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Theorem eleq12 2180
Description: Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
Assertion
Ref Expression
eleq12  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  e.  C  <->  B  e.  D ) )

Proof of Theorem eleq12
StepHypRef Expression
1 eleq1 2178 . 2  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
2 eleq2 2179 . 2  |-  ( C  =  D  ->  ( B  e.  C  <->  B  e.  D ) )
31, 2sylan9bb 455 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  e.  C  <->  B  e.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463
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 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-cleq 2108  df-clel 2111
This theorem is referenced by:  trel  4001  pwnss  4051  epelg  4180  preleq  4438  acexmid  5739  cldval  12163
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