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Theorem eleq12 2242
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 2240 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
2 eleq2 2241 . 2 (𝐶 = 𝐷 → (𝐵𝐶𝐵𝐷))
31, 2sylan9bb 462 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173
This theorem is referenced by:  trel  4105  pwnss  4156  epelg  4286  preleq  4550  acexmid  5867  cldval  13232
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