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Theorem syl5eqelr 2125
 Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eqelr.1 𝐵 = 𝐴
syl5eqelr.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
syl5eqelr (𝜑𝐴𝐶)

Proof of Theorem syl5eqelr
StepHypRef Expression
1 syl5eqelr.1 . . 3 𝐵 = 𝐴
21eqcomi 2044 . 2 𝐴 = 𝐵
3 syl5eqelr.2 . 2 (𝜑𝐵𝐶)
42, 3syl5eqel 2124 1 (𝜑𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ 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:  dmrnssfld  4595  cnvexg  4855  opabbrex  5549  offval  5719  resfunexgALT  5737  abrexexg  5745  abrexex2g  5747  opabex3d  5748  nqprlu  6645  iccshftr  8862  iccshftl  8864  iccdil  8866  icccntr  8868
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