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Theorem abid2f 2777
Description: A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)
Hypothesis
Ref Expression
abid2f.1 𝑥𝐴
Assertion
Ref Expression
abid2f {𝑥𝑥𝐴} = 𝐴

Proof of Theorem abid2f
StepHypRef Expression
1 nfab1 2753 . . 3 𝑥{𝑥𝑥𝐴}
2 abid2f.1 . . 3 𝑥𝐴
31, 2cleqf 2776 . 2 ({𝑥𝑥𝐴} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴))
4 abid 2598 . 2 (𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴)
53, 4mpgbir 1717 1 {𝑥𝑥𝐴} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 195   = wceq 1475  wcel 1977  {cab 2596  wnfc 2738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740
This theorem is referenced by:  mptctf  28883  rabexgf  38206  ssabf  38308  abssf  38326
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