Theorem raleqtrrdv

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

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

Proof

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