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Theorem syl5eqelr 2167
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eqelr.1  |-  B  =  A
syl5eqelr.2  |-  ( ph  ->  B  e.  C )
Assertion
Ref Expression
syl5eqelr  |-  ( ph  ->  A  e.  C )

Proof of Theorem syl5eqelr
StepHypRef Expression
1 syl5eqelr.1 . . 3  |-  B  =  A
21eqcomi 2086 . 2  |-  A  =  B
3 syl5eqelr.2 . 2  |-  ( ph  ->  B  e.  C )
42, 3syl5eqel 2166 1  |-  ( ph  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078
This theorem is referenced by:  dmrnssfld  4623  cnvexg  4885  opabbrex  5580  offval  5750  resfunexgALT  5768  abrexexg  5776  abrexex2g  5778  opabex3d  5779  unfidisj  6442  nqprlu  6799  iccshftr  9092  iccshftl  9094  iccdil  9096  icccntr  9098  exprmfct  10663
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