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, 17-Nov-2019)

	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		nfab1	$$⊢\underset{\_}{Ⅎ}x\left\{x|x\in A\right\}$$
3	2 1	cleqf	$$⊢\left\{x|x\in A\right\}=A\leftrightarrow \forall x\left(x\in \left\{x|x\in A\right\}\leftrightarrow x\in A\right)$$
4		abid	$$⊢x\in \left\{x|x\in A\right\}\leftrightarrow x\in A$$
5	3 4	mpgbir	$$⊢\left\{x|x\in A\right\}=A$$