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Theorem eqabrd 2884
Description: Equality of a class variable and a class abstraction (deduction form of eqabb 2881). (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 2827 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝜓}))
3 abid 2718 . 2 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
42, 3bitrdi 287 1 (𝜑 → (𝑥𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  {cab 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816
This theorem is referenced by:  eqabri  2885  fvelimab  6981  mapsnend  9076  nosupbnd2  27761  noinfbnd2  27776  fvineqsneu  37412  fvineqsneq  37413  ispridlc  38077  ac6s6  38179  dib1dim  41167  prprspr2  47505
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