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Theorem syl6eqelr 2129
 Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl6eqelr.1 (𝜑𝐵 = 𝐴)
syl6eqelr.2 𝐵𝐶
Assertion
Ref Expression
syl6eqelr (𝜑𝐴𝐶)

Proof of Theorem syl6eqelr
StepHypRef Expression
1 syl6eqelr.1 . . 3 (𝜑𝐵 = 𝐴)
21eqcomd 2045 . 2 (𝜑𝐴 = 𝐵)
3 syl6eqelr.2 . 2 𝐵𝐶
42, 3syl6eqel 2128 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:  eusvnfb  4186  releldm2  5811  bren  6228  brdomg  6229  ioof  8840
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