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Theorem abid2f 2987
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 2960 . . 3 𝑥{𝑥𝑥𝐴}
2 abid2f.1 . . 3 𝑥𝐴
31, 2cleqf 2986 . 2 ({𝑥𝑥𝐴} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴))
4 abid 2783 . 2 (𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴)
53, 4mpgbir 1801 1 {𝑥𝑥𝐴} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2112  {cab 2779  wnfc 2939
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941
This theorem is referenced by:  mptctf  30482  rabexgf  41640  ssabf  41723  abssf  41735
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