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Theorem abid2f 2942
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 2941 . . 3 (𝐴 = {𝑥𝑥𝐴} ↔ ∀𝑥(𝑥𝐴𝑥𝐴))
3 biid 261 . . 3 (𝑥𝐴𝑥𝐴)
42, 3mpgbir 1797 . 2 𝐴 = {𝑥𝑥𝐴}
54eqcomi 2749 1 {𝑥𝑥𝐴} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2108  {cab 2717  wnfc 2893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895
This theorem is referenced by:  mptctf  32733  rabexgf  44926  ssabf  45004  abssf  45016
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