Theorem raleqtrdv

Description: Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025)

	Ref	Expression
Hypotheses	raleqtrdv.1	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )
	raleqtrdv.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	raleqtrdv	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		raleqtrdv.1	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )
2		raleqtrdv.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3	2	raleqdv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) )
4	1 3	mpbid	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 𝜓 )