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Theorem syl6eqelr 2180
 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 2094 . 2 (𝜑𝐴 = 𝐵)
3 syl6eqelr.2 . 2 𝐵𝐶
42, 3syl6eqel 2179 1 (𝜑𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1290   ∈ wcel 1439 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-17 1465  ax-ial 1473  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-cleq 2082  df-clel 2085 This theorem is referenced by:  eusvnfb  4291  releldm2  5971  mapprc  6425  ixpprc  6492  ixpssmap2g  6500  ixpssmapg  6501  bren  6520  brdomg  6521  mapen  6618  ssenen  6623  ioof  9452  hashfacen  10304  fisum  10841
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