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Theorem eleq12i 2183
Description: Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
Hypotheses
Ref Expression
eleq1i.1  |-  A  =  B
eleq12i.2  |-  C  =  D
Assertion
Ref Expression
eleq12i  |-  ( A  e.  C  <->  B  e.  D )

Proof of Theorem eleq12i
StepHypRef Expression
1 eleq12i.2 . . 3  |-  C  =  D
21eleq2i 2182 . 2  |-  ( A  e.  C  <->  A  e.  D )
3 eleq1i.1 . . 3  |-  A  =  B
43eleq1i 2181 . 2  |-  ( A  e.  D  <->  B  e.  D )
52, 4bitri 183 1  |-  ( A  e.  C  <->  B  e.  D )
Colors of variables: wff set class
Syntax hints:    <-> 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:  3eltr3g  2200  3eltr4g  2201  sbcel12g  2986  ennnfonelem1  11815
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