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Theorem eleq12 2345
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 2343 . 2  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
2 eleq2 2344 . 2  |-  ( C  =  D  ->  ( B  e.  C  <->  B  e.  D ) )
31, 2sylan9bb 680 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  e.  C  <->  B  e.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684
This theorem is referenced by:  trel  4120  pwnss  4176  epelg  4306  preleq  7318  oemapval  7385  cantnf  7395  wemapwe  7400  nnsdomel  7623  cldval  16760  isufil  17598  issiga  23472  wepwsolem  27138  aomclem8  27159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-cleq 2276  df-clel 2279
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