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	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	abid2f	$$⊢\left\{x|x\in A\right\}=A$$

Proof

Step	Hyp	Ref	Expression
1		abid2f.1	$$⊢\underset{\_}{Ⅎ}xA$$
2	1	eqabf	$$⊢A=\left\{x|x\in A\right\}\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in A\right)$$
3		biid	$$⊢x\in A\leftrightarrow x\in A$$
4	2 3	mpgbir	$$⊢A=\left\{x|x\in A\right\}$$
5	4	eqcomi	$$⊢\left\{x|x\in A\right\}=A$$