Theorem eqrdv

Description: Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996)

	Ref	Expression
Hypothesis	eqrdv.1	$$⊢\phi \to \left(x\in A\leftrightarrow x\in B\right)$$
Assertion	eqrdv	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		eqrdv.1	$$⊢\phi \to \left(x\in A\leftrightarrow x\in B\right)$$
2	1	alrimiv	$$⊢\phi \to \forall x\left(x\in A\leftrightarrow x\in B\right)$$
3		dfcleq	$$⊢A=B\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in B\right)$$
4	2 3	sylibr	$$⊢\phi \to A=B$$