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Theorem syl6eqelr 2176
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 2090 . 2 (𝜑𝐴 = 𝐵)
3 syl6eqelr.2 . 2 𝐵𝐶
42, 3syl6eqel 2175 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  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:  eusvnfb  4243  releldm2  5893  mapprc  6342  bren  6397  brdomg  6398  ctex  6403  mapen  6495  ssenen  6500  ioof  9298  hashfacen  10089
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