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Theorem eleq12 2831
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 2829 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
2 eleq2 2830 . 2 (𝐶 = 𝐷 → (𝐵𝐶𝐵𝐷))
31, 2sylan9bb 509 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-clel 2816
This theorem is referenced by:  rru  3785  trel  5268  epelg  5585  preleqg  9655  preleqALT  9657  oemapval  9723  cantnf  9733  wemapwe  9737  nnsdomel  10030  cldval  23031  isufil  23911  taylthlem2  26416  umgr2v2enb1  29544  issiga  34113  bj-epelg  37069  rdgssun  37379  matunitlindf  37625  wepwsolem  43054  aomclem8  43073  grumnud  44305  nelbr  47286
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