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Theorem eqabrd 2868
Description: Equality of a class variable and a class abstraction (deduction form of eqabb 2865). (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 2811 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝜓}))
3 abid 2705 . 2 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
42, 3bitrdi 287 1 (𝜑 → (𝑥𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  {cab 2701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802
This theorem is referenced by:  eqabri  2869  fvelimab  6955  mapsnend  9033  nosupbnd2  27590  noinfbnd2  27605  fvineqsneu  36793  fvineqsneq  36794  ispridlc  37442  ac6s6  37544  dib1dim  40540  prprspr2  46732
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