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Theorem abid2f 2820
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 2795 . . 3 𝑥{𝑥𝑥𝐴}
2 abid2f.1 . . 3 𝑥𝐴
31, 2cleqf 2819 . 2 ({𝑥𝑥𝐴} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴))
4 abid 2639 . 2 (𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴)
53, 4mpgbir 1766 1 {𝑥𝑥𝐴} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wcel 2030  {cab 2637  wnfc 2780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782
This theorem is referenced by:  mptctf  29623  rabexgf  39497  ssabf  39594  abssf  39609
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