Theorem eqrd

Description: Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017) (Proof shortened by BJ, 1-Dec-2021)

	Ref	Expression
Hypotheses	eqrd.0	$$⊢Ⅎx\phi$$
	eqrd.1	$$⊢\underset{\_}{Ⅎ}xA$$
	eqrd.2	$$⊢\underset{\_}{Ⅎ}xB$$
	eqrd.3	$$⊢\phi \to \left(x\in A\leftrightarrow x\in B\right)$$
Assertion	eqrd	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		eqrd.0	$$⊢Ⅎx\phi$$
2		eqrd.1	$$⊢\underset{\_}{Ⅎ}xA$$
3		eqrd.2	$$⊢\underset{\_}{Ⅎ}xB$$
4		eqrd.3	$$⊢\phi \to \left(x\in A\leftrightarrow x\in B\right)$$
5	1 4	alrimi	$$⊢\phi \to \forall x\left(x\in A\leftrightarrow x\in B\right)$$
6	2 3	cleqf	$$⊢A=B\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in B\right)$$
7	5 6	sylibr	$$⊢\phi \to A=B$$