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Theorem eleqtrd 2116
Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
Hypotheses
Ref Expression
eleqtrd.1  |-  ( ph  ->  A  e.  B )
eleqtrd.2  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
eleqtrd  |-  ( ph  ->  A  e.  C )

Proof of Theorem eleqtrd
StepHypRef Expression
1 eleqtrd.1 . 2  |-  ( ph  ->  A  e.  B )
2 eleqtrd.2 . . 3  |-  ( ph  ->  B  =  C )
32eleq2d 2107 . 2  |-  ( ph  ->  ( A  e.  B  <->  A  e.  C ) )
41, 3mpbid 135 1  |-  ( ph  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036
This theorem is referenced by:  eleqtrrd  2117  3eltr3d  2120  syl5eleq  2126  syl6eleq  2130  opth1  3973  0nelop  3985  tfisi  4310  ercl  6117  erth  6150  ecelqsdm  6176  phpm  6327  lincmb01cmp  8869  fzopth  8922  fzoaddel2  9047  fzosubel2  9049  fzocatel  9053  zpnn0elfzo1  9062  fzoend  9076  peano2fzor  9086  monoord2  9210  isermono  9211
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