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Theorem abid2f 2986
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 2961 . . 3 𝑥{𝑥𝑥𝐴}
2 abid2f.1 . . 3 𝑥𝐴
31, 2cleqf 2985 . 2 ({𝑥𝑥𝐴} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴))
4 abid 2805 . 2 (𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴)
53, 4mpgbir 1881 1 {𝑥𝑥𝐴} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1637  wcel 2157  {cab 2803  wnfc 2946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948
This theorem is referenced by:  mptctf  29845  rabexgf  39695  ssabf  39791  abssf  39805
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