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Theorem eleq1a 2125
Description: A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
Assertion
Ref Expression
eleq1a  |-  ( A  e.  B  ->  ( C  =  A  ->  C  e.  B ) )

Proof of Theorem eleq1a
StepHypRef Expression
1 eleq1 2116 . 2  |-  ( C  =  A  ->  ( C  e.  B  <->  A  e.  B ) )
21biimprcd 153 1  |-  ( A  e.  B  ->  ( C  =  A  ->  C  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259    e. wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052
This theorem is referenced by:  elex22  2586  elex2  2587  reu6  2753  disjne  3301  ssimaex  5262  fnex  5411  f1ocnv2d  5732  tfrlem8  5965  eroprf  6230  ac6sfi  6383  recclnq  6548  prnmaddl  6646  renegcl  7335  nn0ind-raph  8414  iccid  8895  bj-nn0suc  10476  bj-inf2vnlem2  10483  bj-nn0sucALT  10490
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