Theorem raleq

Description: Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023)

	Ref	Expression
Assertion	raleq	⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		biidd	⊢ ( 𝐴 = 𝐵 → ( 𝜑 ↔ 𝜑 ) )
2	1	raleqbi1dv	⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 𝜑 ) )