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Theorem syl5eleq 2706
Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eleq.1 𝐴𝐵
syl5eleq.2 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
syl5eleq (𝜑𝐴𝐶)

Proof of Theorem syl5eleq
StepHypRef Expression
1 syl5eleq.1 . . 3 𝐴𝐵
21a1i 11 . 2 (𝜑𝐴𝐵)
3 syl5eleq.2 . 2 (𝜑𝐵 = 𝐶)
42, 3eleqtrd 2702 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704  df-cleq 2614  df-clel 2617
This theorem is referenced by:  syl5eleqr  2707  opth1  4942  opth  4943  eqelsuc  5804  tfrlem11  7481  oalimcl  7637  omlimcl  7655  frgp0  18167  txdis  21429  ordtconnlem1  29955  rankeq1o  32262
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