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Theorem eleqtrrd 2162
Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
Hypotheses
Ref Expression
eleqtrrd.1  |-  ( ph  ->  A  e.  B )
eleqtrrd.2  |-  ( ph  ->  C  =  B )
Assertion
Ref Expression
eleqtrrd  |-  ( ph  ->  A  e.  C )

Proof of Theorem eleqtrrd
StepHypRef Expression
1 eleqtrrd.1 . 2  |-  ( ph  ->  A  e.  B )
2 eleqtrrd.2 . . 3  |-  ( ph  ->  C  =  B )
32eqcomd 2088 . 2  |-  ( ph  ->  B  =  C )
41, 3eleqtrd 2161 1  |-  ( ph  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-clel 2079
This theorem is referenced by:  3eltr4d  2166  tfrexlem  6031  nnsucuniel  6188  erref  6242  en1uniel  6451  fin0  6531  fin0or  6532  prarloclemarch2  6881  fzopth  9369  fzoss2  9472  fzosubel3  9496  elfzomin  9506  elfzonlteqm1  9510  fzoend  9522  fzofzp1  9527  fzofzp1b  9528  peano2fzor  9532  zmodfzo  9643  frecuzrdg0  9709  frecuzrdgsuc  9710  frecuzrdgdomlem  9713  frecuzrdg0t  9718  frecuzrdgsuctlem  9719  bcn2  10007
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