Theorem raleqdv

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005)

	Ref	Expression
Hypothesis	raleq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	raleqdv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		raleq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		raleq	⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) )
3	1 2	syl	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) )