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Theorem eleqtrid 2226
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
eleqtrid.1  |-  A  e.  B
eleqtrid.2  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
eleqtrid  |-  ( ph  ->  A  e.  C )

Proof of Theorem eleqtrid
StepHypRef Expression
1 eleqtrid.1 . . 3  |-  A  e.  B
21a1i 9 . 2  |-  ( ph  ->  A  e.  B )
3 eleqtrid.2 . 2  |-  ( ph  ->  B  =  C )
42, 3eleqtrd 2216 1  |-  ( ph  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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:  eleqtrrid  2227  opth1  4153  opth  4154  eqelsuc  4336  txdis  12435  bj-nnelirr  13140
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