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	⊢ 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 2934	. . 3 ⊢ (𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
3		biid 261	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
4	2, 3	mpgbir 1800	. 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
5	4	eqcomi 2740	1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Syntax hints:   ↔ wb 205   = wceq 1540   ∈ wcel 2105  {cab 2708  Ⅎwnfc 2882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884

This theorem is referenced by:  mptctf  32375  rabexgf  44171  ssabf  44251  abssf  44263