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Theorem eleq12 2879
 Description: Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
Assertion
Ref Expression
eleq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))

Proof of Theorem eleq12
StepHypRef Expression
1 eleq1 2877 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
2 eleq2 2878 . 2 (𝐶 = 𝐷 → (𝐵𝐶𝐵𝐷))
31, 2sylan9bb 513 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870 This theorem is referenced by:  rru  3718  trel  5143  epelg  5431  preleqg  9062  preleqALT  9064  oemapval  9130  cantnf  9140  wemapwe  9144  nnsdomel  9403  cldval  21628  isufil  22508  umgr2v2enb1  27316  issiga  31481  bj-epelg  34484  rdgssun  34795  fvineqsneu  34828  matunitlindf  35055  wepwsolem  39984  aomclem8  40003  grumnud  40992  nelbr  43828
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