Theorem abeq1i

Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994) (Proof shortened by Wolf Lammen, 15-Nov-2019)

	Ref	Expression
Hypothesis	abeq1i.1	$$⊢\left\{x|\phi \right\}=A$$
Assertion	abeq1i	$$⊢\phi \leftrightarrow x\in A$$

Proof

Step	Hyp	Ref	Expression
1		abeq1i.1	$$⊢\left\{x|\phi \right\}=A$$
2	1	eqcomi	$$⊢A=\left\{x|\phi \right\}$$
3	2	abeq2i	$$⊢x\in A\leftrightarrow \phi$$
4	3	bicomi	$$⊢\phi \leftrightarrow x\in A$$