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Theorem eleq12i 2681
 Description: Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
Hypotheses
Ref Expression
eleq1i.1 𝐴 = 𝐵
eleq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
eleq12i (𝐴𝐶𝐵𝐷)

Proof of Theorem eleq12i
StepHypRef Expression
1 eleq12i.2 . . 3 𝐶 = 𝐷
21eleq2i 2680 . 2 (𝐴𝐶𝐴𝐷)
3 eleq1i.1 . . 3 𝐴 = 𝐵
43eleq1i 2679 . 2 (𝐴𝐷𝐵𝐷)
52, 4bitri 263 1 (𝐴𝐶𝐵𝐷)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606 This theorem is referenced by:  sbcel12  3935  zclmncvs  22756  gausslemma2dlem4  24894  bnj98  30191  elmpst  30687  elmpps  30724  sbcel12gOLD  37775  unirnmapsn  38401
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