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Theorem eqabrd 2872
Description: Equality of a class variable and a class abstraction (deduction form of eqabb 2869). (Contributed by NM, 16-Nov-1995.)
Hypothesis
Ref Expression
eqabrd.1 (𝜑𝐴 = {𝑥𝜓})
Assertion
Ref Expression
eqabrd (𝜑 → (𝑥𝐴𝜓))

Proof of Theorem eqabrd
StepHypRef Expression
1 eqabrd.1 . . 3 (𝜑𝐴 = {𝑥𝜓})
21eleq2d 2815 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝜓}))
3 abid 2709 . 2 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
42, 3bitrdi 287 1 (𝜑 → (𝑥𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  {cab 2705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806
This theorem is referenced by:  eqabri  2873  fvelimab  6971  mapsnend  9061  nosupbnd2  27662  noinfbnd2  27677  fvineqsneu  36890  fvineqsneq  36891  ispridlc  37543  ac6s6  37645  dib1dim  40638  prprspr2  46858
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