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Theorem syl5eleq 2693
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 2689 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-cleq 2602  df-clel 2605
This theorem is referenced by:  syl5eleqr  2694  opth1  4863  opth  4864  eqelsuc  5708  tfrlem11  7348  oalimcl  7504  omlimcl  7522  frgp0  17944  txdis  21192  ordtconlem1  29091  rankeq1o  31241
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