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Metamath Proof Explorer

Theorem eqriv

Description: Infer equality of classes from equivalence of membership. (Contributed by NM, 21-Jun-1993)

		Ref	Expression
	Hypothesis	eqriv.1	$⊢ x \in A \leftrightarrow x \in B$
	Assertion	eqriv	$⊢ A = B$

Step	Hyp	Ref	Expression
1		eqriv.1	$⊢ x \in A \leftrightarrow x \in B$
2		dfcleq	$⊢ A = B \leftrightarrow \forall x (x \in A \leftrightarrow x \in B)$
3	2 1	mpgbir	$⊢ A = B$