Theorem eqriv

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

	Ref	Expression
Hypothesis	eqriv.1	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
Assertion	eqriv	⊢ 𝐴 = 𝐵

Proof

Step	Hyp	Ref	Expression
1		eqriv.1	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
2		dfcleq	⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
3	2 1	mpgbir	⊢ 𝐴 = 𝐵