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Theorem eleqtri 2212
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 2204 . 2  |-  ( A  e.  B  <->  A  e.  C )
41, 3mpbi 144 1  |-  A  e.  C
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-clel 2133
This theorem is referenced by:  eleqtrri  2213  3eltr3i  2218  prid2  3625  2eluzge0  9363  fz01or  9884  ef0lem  11355  ege2le3  11366  efgt1p2  11390  efgt1p  11391  phi1  11884  cnrehmeocntop  12751  dvcjbr  12830
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