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Theorem eleq12 2828
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 2826 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
2 eleq2 2827 . 2 (𝐶 = 𝐷 → (𝐵𝐶𝐵𝐷))
31, 2sylan9bb 511 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2729  df-clel 2815
This theorem is referenced by:  rru  3742  trel  5236  epelg  5543  preleqg  9558  preleqALT  9560  oemapval  9626  cantnf  9636  wemapwe  9640  nnsdomel  9933  cldval  22390  isufil  23270  umgr2v2enb1  28516  issiga  32751  bj-epelg  35568  rdgssun  35878  fvineqsneu  35911  matunitlindf  36105  wepwsolem  41398  aomclem8  41417  grumnud  42640  nelbr  45580
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