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Theorem eleq12i 2155
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 2154 . 2  |-  ( A  e.  C  <->  A  e.  D )
3 eleq1i.1 . . 3  |-  A  =  B
43eleq1i 2153 . 2  |-  ( A  e.  D  <->  B  e.  D )
52, 4bitri 182 1  |-  ( A  e.  C  <->  B  e.  D )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1289    e. wcel 1438
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084
This theorem is referenced by:  3eltr3g  2172  3eltr4g  2173  sbcel12g  2944
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