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Theorem eqeltrrdi 2231
 Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
eqeltrrdi.1 (𝜑𝐵 = 𝐴)
eqeltrrdi.2 𝐵𝐶
Assertion
Ref Expression
eqeltrrdi (𝜑𝐴𝐶)

Proof of Theorem eqeltrrdi
StepHypRef Expression
1 eqeltrrdi.1 . . 3 (𝜑𝐵 = 𝐴)
21eqcomd 2145 . 2 (𝜑𝐴 = 𝐵)
3 eqeltrrdi.2 . 2 𝐵𝐶
42, 3eqeltrdi 2230 1 (𝜑𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135 This theorem is referenced by:  eusvnfb  4375  releldm2  6083  mapprc  6546  ixpprc  6613  ixpssmap2g  6621  ixpssmapg  6622  bren  6641  brdomg  6642  mapen  6740  ssenen  6745  fi0  6863  ioof  9766  hashfacen  10591  fsum3  11168  cnrehmeocntop  12776
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