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Theorem abid2f 2935
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, 26-Feb-2025.)
Hypothesis
Ref Expression
abid2f.1 𝑥𝐴
Assertion
Ref Expression
abid2f {𝑥𝑥𝐴} = 𝐴

Proof of Theorem abid2f
StepHypRef Expression
1 abid2f.1 . . . 4 𝑥𝐴
21eqabf 2934 . . 3 (𝐴 = {𝑥𝑥𝐴} ↔ ∀𝑥(𝑥𝐴𝑥𝐴))
3 biid 260 . . 3 (𝑥𝐴𝑥𝐴)
42, 3mpgbir 1801 . 2 𝐴 = {𝑥𝑥𝐴}
54eqcomi 2740 1 {𝑥𝑥𝐴} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wcel 2106  {cab 2708  wnfc 2882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884
This theorem is referenced by:  mptctf  31808  rabexgf  43465  ssabf  43546  abssf  43558
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