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Theorem syl5eleq 2855
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 2851 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-cleq 2763  df-clel 2766
This theorem is referenced by:  syl5eleqr  2856  opth1  5071  opth  5072  eqelsuc  5949  tfrlem11  7636  oalimcl  7793  omlimcl  7811  frgp0  18379  txdis  21655  ordtconnlem1  30304  rankeq1o  32609
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