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Theorem eleq12i 2245
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 2244 . 2  |-  ( A  e.  C  <->  A  e.  D )
3 eleq1i.1 . . 3  |-  A  =  B
43eleq1i 2243 . 2  |-  ( A  e.  D  <->  B  e.  D )
52, 4bitri 184 1  |-  ( A  e.  C  <->  B  e.  D )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1353    e. wcel 2148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173
This theorem is referenced by:  3eltr3g  2262  3eltr4g  2263  sbcel12g  3072  ennnfonelem1  12378
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