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Theorem eleq12 2182
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 2180 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
2 eleq2 2181 . 2 (𝐶 = 𝐷 → (𝐵𝐶𝐵𝐷))
31, 2sylan9bb 457 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-clel 2113
This theorem is referenced by:  trel  4003  pwnss  4053  epelg  4182  preleq  4440  acexmid  5741  cldval  12195
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