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Theorem eleq12 2852
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 2850 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
2 eleq2 2851 . 2 (𝐶 = 𝐷 → (𝐵𝐶𝐵𝐷))
31, 2sylan9bb 518 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  wb 209  wa 400   = wceq 1569  wcel 2142
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734
This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-cleq 2754  df-clel 2837
This theorem is used by:  rru  3741  trel  5225  epelg  5561  preleqg  9582  preleqALT  9584  oemapval  9650  cantnf  9660  wemapwe  9664  nnsdomel  9983  cldval  23191  isufil  24071  taylthlem2  26548  umgr2v2enb1  29887  issiga  34511  bj-epelg  37732  rdgssun  38052  matunitlindf  38297  wepwsolem  43797  aomclem8  43816  grumnud  45024  nelbr  48039
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