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Theorem eleq12 2497
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 2495 . 2  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
2 eleq2 2496 . 2  |-  ( C  =  D  ->  ( B  e.  C  <->  B  e.  D ) )
31, 2sylan9bb 681 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  e.  C  <->  B  e.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725
This theorem is referenced by:  trel  4301  pwnss  4357  epelg  4487  preleq  7564  oemapval  7631  cantnf  7641  wemapwe  7646  nnsdomel  7869  cldval  17079  isufil  17927  issiga  24486  wepwsolem  27107  aomclem8  27127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-cleq 2428  df-clel 2431
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