Theorem eqabi

Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 26-May-1993) Avoid ax-11 . (Revised by Wolf Lammen, 6-May-2023)

	Ref	Expression
Hypothesis	eqabi.1	$$⊢x\in A\leftrightarrow \phi$$
Assertion	eqabi	$$⊢A=\left\{x|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		eqabi.1	$$⊢x\in A\leftrightarrow \phi$$
2	1	a1i	$$⊢\top \to \left(x\in A\leftrightarrow \phi \right)$$
3	2	eqabdv	$$⊢\top \to A=\left\{x|\phi \right\}$$
4	3	mptru	$$⊢A=\left\{x|\phi \right\}$$