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Theorem eleq12i 2902
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 2901 . 2 (𝐴𝐶𝐴𝐷)
3 eleq1i.1 . . 3 𝐴 = 𝐵
43eleq1i 2900 . 2 (𝐴𝐷𝐵𝐷)
52, 4bitri 276 1 (𝐴𝐶𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2811  df-clel 2890
This theorem is referenced by:  sbcel12  4357  zclmncvs  23679  gausslemma2dlem4  25872  bnj98  32038  elmpst  32680  elmpps  32717  unirnmapsn  41353  smndex1n0mnd  44012
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