Theorem abid2f 2962

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	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
abid2f	⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Proof of Theorem abid2f

Step	Hyp	Ref	Expression
1		abid2f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	eqabf 2961	. . 3 ⊢ (𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
3		biid 264	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
4	2, 3	mpgbir 1827	. 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
5	4	eqcomi 2779	1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Syntax hints:   ↔ wb 209   = wceq 1568   ∈ wcel 2150  {cab 2748  Ⅎwnfc 2917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919

This theorem is referenced by:  mptctf  33031  rabexgf  45696  ssabf  45770  abssf  45782