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Theorem eleq12 2099
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 2097 . 2 (A = B → (A 𝐶B 𝐶))
2 eleq2 2098 . 2 (𝐶 = 𝐷 → (B 𝐶B 𝐷))
31, 2sylan9bb 435 1 ((A = B 𝐶 = 𝐷) → (A 𝐶B 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390
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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033
This theorem is referenced by:  trel  3852  pwnss  3903  epelg  4018  preleq  4233  acexmid  5454
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