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Theorem eleqtri 2192
Description: Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
eleqtr.1  |-  A  e.  B
eleqtr.2  |-  B  =  C
Assertion
Ref Expression
eleqtri  |-  A  e.  C

Proof of Theorem eleqtri
StepHypRef Expression
1 eleqtr.1 . 2  |-  A  e.  B
2 eleqtr.2 . . 3  |-  B  =  C
32eleq2i 2184 . 2  |-  ( A  e.  B  <->  A  e.  C )
41, 3mpbi 144 1  |-  A  e.  C
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-clel 2113
This theorem is referenced by:  eleqtrri  2193  3eltr3i  2198  prid2  3600  2eluzge0  9338  fz01or  9859  ef0lem  11293  ege2le3  11304  efgt1p2  11328  efgt1p  11329  phi1  11822  cnrehmeocntop  12689  dvcjbr  12768
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