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Theorem eleq12 2519
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 2517 . 2  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
2 eleq2 2518 . 2  |-  ( C  =  D  ->  ( B  e.  C  <->  B  e.  D ) )
31, 2sylan9bb 706 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  e.  C  <->  B  e.  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-cleq 2444  df-clel 2447
This theorem is referenced by:  trel  4504  pwnss  4568  epelg  4746  preleq  8122  oemapval  8188  cantnf  8198  wemapwe  8202  nnsdomel  8424  cldval  20038  isufil  20918  issiga  28933  wepwsolem  35900  aomclem8  35919  umgr2v2enb1  39563
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