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Theorem syl5eqelr 2172
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 2089 . 2  |-  A  =  B
3 syl5eqelr.2 . 2  |-  ( ph  ->  B  e.  C )
42, 3syl5eqel 2171 1  |-  ( ph  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    e. wcel 1436
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-cleq 2078  df-clel 2081
This theorem is referenced by:  dmrnssfld  4666  cnvexg  4936  opabbrex  5652  offval  5822  resfunexgALT  5840  abrexexg  5848  abrexex2g  5850  opabex3d  5851  unfidisj  6586  nqprlu  7053  iccshftr  9346  iccshftl  9348  iccdil  9350  icccntr  9352  exprmfct  11025
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