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Theorem eleq12 2819
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 2817 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
2 eleq2 2818 . 2 (𝐶 = 𝐷 → (𝐵𝐶𝐵𝐷))
31, 2sylan9bb 509 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-cleq 2720  df-clel 2806
This theorem is referenced by:  rru  3774  trel  5274  epelg  5583  preleqg  9639  preleqALT  9641  oemapval  9707  cantnf  9717  wemapwe  9721  nnsdomel  10014  cldval  22940  isufil  23820  taylthlem2  26322  umgr2v2enb1  29353  issiga  33731  bj-epelg  36547  rdgssun  36857  fvineqsneu  36890  matunitlindf  37091  wepwsolem  42466  aomclem8  42485  grumnud  43723  nelbr  46654
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