Theorem eqrdv

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

	Ref	Expression
Hypothesis	eqrdv.1	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
Assertion	eqrdv	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		eqrdv.1	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
2	1	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
3		dfcleq	⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
4	2 3	sylibr	⊢ ( 𝜑 → 𝐴 = 𝐵 )