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Theorem eleqtrid 2922
Description: A 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 11 . 2 (𝜑𝐴𝐵)
3 eleqtrid.2 . 2 (𝜑𝐵 = 𝐶)
42, 3eleqtrd 2918 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817  df-clel 2896
This theorem is referenced by:  eleqtrrid  2923  opth1  5355  opth  5356  eqelsuc  6260  tfrlem11  8016  oalimcl  8178  omlimcl  8196  frgp0  18884  txdis  22235  ordtconnlem1  31194  rankeq1o  33659
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