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Theorem eleq12i 2832
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 2831 . 2 (𝐴𝐶𝐴𝐷)
3 eleq1i.1 . . 3 𝐴 = 𝐵
43eleq1i 2830 . 2 (𝐴𝐷𝐵𝐷)
52, 4bitri 275 1 (𝐴𝐶𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-clel 2814
This theorem is referenced by:  sbcel12  4417  smndex1n0mnd  18938  zclmncvs  25196  gausslemma2dlem4  27428  bnj98  34860  elmpst  35521  elmpps  35558  sbceqbii  36173  cbvsbcvw2  36213  oaordnrex  43285  omnord1ex  43294  oenord1ex  43305  unirnmapsn  45157
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