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Theorem syl5eleq 2736
 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 2732 1 (𝜑𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-cleq 2644  df-clel 2647 This theorem is referenced by:  syl5eleqr  2737  opth1  4973  opth  4974  eqelsuc  5844  tfrlem11  7529  oalimcl  7685  omlimcl  7703  frgp0  18219  txdis  21483  ordtconnlem1  30098  rankeq1o  32403
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