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Theorem eleq12 2144
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 2142 . 2  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
2 eleq2 2143 . 2  |-  ( C  =  D  ->  ( B  e.  C  <->  B  e.  D ) )
31, 2sylan9bb 450 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  e.  C  <->  B  e.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078
This theorem is referenced by:  trel  3884  pwnss  3935  epelg  4047  preleq  4300  acexmid  5536
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