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Theorem syl5eleq 2126
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eleq.1  |-  A  e.  B
syl5eleq.2  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
syl5eleq  |-  ( ph  ->  A  e.  C )

Proof of Theorem syl5eleq
StepHypRef Expression
1 syl5eleq.1 . . 3  |-  A  e.  B
21a1i 9 . 2  |-  ( ph  ->  A  e.  B )
3 syl5eleq.2 . 2  |-  ( ph  ->  B  =  C )
42, 3eleqtrd 2116 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:  syl5eleqr  2127  opth1  3973  opth  3974  eqelsuc  4156  bj-nnelirr  10078
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