Theorem cbvralvw2

Description: Change bound variable and domain in the restricted universal quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvralvw2.1	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
	cbvralvw2.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvralvw2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvralvw2.1	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
2		cbvralvw2.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
4	1	eleq2d	⊢ ( 𝑥 = 𝑦 → ( 𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
5	3 4	bitrd	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
6	5 2	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑦 ∈ 𝐵 → 𝜓 ) ) )
7	6	cbvalvw	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜓 ) )
8		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
9		df-ral	⊢ ( ∀ 𝑦 ∈ 𝐵 𝜓 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜓 ) )
10	7 8 9	3bitr4i	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 𝜓 )