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Theorem eqabrd 2882
Description: Equality of a class variable and a class abstraction (deduction form of eqabb 2879). (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 2825 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝜓}))
3 abid 2716 . 2 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
42, 3bitrdi 287 1 (𝜑 → (𝑥𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  {cab 2712
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-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814
This theorem is referenced by:  eqabri  2883  fvelimab  6981  mapsnend  9075  nosupbnd2  27776  noinfbnd2  27791  fvineqsneu  37394  fvineqsneq  37395  ispridlc  38057  ac6s6  38159  dib1dim  41148  prprspr2  47443
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