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Theorem eleq12 2874
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 2872 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
2 eleq2 2873 . 2 (𝐶 = 𝐷 → (𝐵𝐶𝐵𝐷))
31, 2sylan9bb 510 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wcel 2083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1766  df-cleq 2790  df-clel 2865
This theorem is referenced by:  rru  3709  trel  5077  epelg  5361  epelgOLD  5362  preleqg  8931  preleqALT  8933  oemapval  8999  cantnf  9009  wemapwe  9013  nnsdomel  9272  cldval  21319  isufil  22199  umgr2v2enb1  26995  issiga  30984  bj-elep  33979  rdgssun  34211  fvineqsneu  34244  matunitlindf  34442  wepwsolem  39148  aomclem8  39167  grumnud  40140  nelbr  43011
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