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Theorem eleq12 2084
Description: Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
Assertion
Ref Expression
eleq12 ((A = B 𝐶 = 𝐷) → (A 𝐶B 𝐷))

Proof of Theorem eleq12
StepHypRef Expression
1 eleq1 2082 . 2 (A = B → (A 𝐶B 𝐶))
2 eleq2 2083 . 2 (𝐶 = 𝐷 → (B 𝐶B 𝐷))
31, 2sylan9bb 438 1 ((A = B 𝐶 = 𝐷) → (A 𝐶B 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-cleq 2015  df-clel 2018
This theorem is referenced by:  trel  3835  pwnss  3886  epelg  4001  preleq  4217  acexmid  5435
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