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Theorem syl5eleqr 2177
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eleqr.1  |-  A  e.  B
syl5eleqr.2  |-  ( ph  ->  C  =  B )
Assertion
Ref Expression
syl5eleqr  |-  ( ph  ->  A  e.  C )

Proof of Theorem syl5eleqr
StepHypRef Expression
1 syl5eleqr.1 . 2  |-  A  e.  B
2 syl5eleqr.2 . . 3  |-  ( ph  ->  C  =  B )
32eqcomd 2093 . 2  |-  ( ph  ->  B  =  C )
41, 3syl5eleq 2176 1  |-  ( ph  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084
This theorem is referenced by:  rabsnt  3512  0elnn  4422  tfrexlem  6081  rdgtfr  6121  rdgruledefgg  6122  hashinfom  10151
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