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Theorem eleqtrid 2228
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
eleqtrid.1 𝐴𝐵
eleqtrid.2 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
eleqtrid (𝜑𝐴𝐶)

Proof of Theorem eleqtrid
StepHypRef Expression
1 eleqtrid.1 . . 3 𝐴𝐵
21a1i 9 . 2 (𝜑𝐴𝐵)
3 eleqtrid.2 . 2 (𝜑𝐵 = 𝐶)
42, 3eleqtrd 2218 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:  eleqtrrid  2229  opth1  4158  opth  4159  eqelsuc  4341  txdis  12460  bj-nnelirr  13256
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