Theorem abid2f

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)

	Ref	Expression
Hypothesis	abid2f.1	⊢ Ⅎ 𝑥 𝐴
Assertion	abid2f	⊢ { 𝑥 ∣ 𝑥 ∈ 𝐴 } = 𝐴

Proof

Step	Hyp	Ref	Expression
1		abid2f.1	⊢ Ⅎ 𝑥 𝐴
2	1	eqabf	⊢ ( 𝐴 = { 𝑥 ∣ 𝑥 ∈ 𝐴 } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴 ) )
3		biid	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴 )
4	2 3	mpgbir	⊢ 𝐴 = { 𝑥 ∣ 𝑥 ∈ 𝐴 }
5	4	eqcomi	⊢ { 𝑥 ∣ 𝑥 ∈ 𝐴 } = 𝐴