Theorem cbvral2vw

Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 10-Aug-2004) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvral2vw.1	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜒 ) )
	cbvral2vw.2	⊢ ( 𝑦 = 𝑤 → ( 𝜒 ↔ 𝜓 ) )
Assertion	cbvral2vw	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑧 ∈ 𝐴 ∀ 𝑤 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvral2vw.1	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜒 ) )
2		cbvral2vw.2	⊢ ( 𝑦 = 𝑤 → ( 𝜒 ↔ 𝜓 ) )
3	1	ralbidv	⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 𝜒 ) )
4	3	cbvralvw	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑧 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 )
5	2	cbvralvw	⊢ ( ∀ 𝑦 ∈ 𝐵 𝜒 ↔ ∀ 𝑤 ∈ 𝐵 𝜓 )
6	5	ralbii	⊢ ( ∀ 𝑧 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 ↔ ∀ 𝑧 ∈ 𝐴 ∀ 𝑤 ∈ 𝐵 𝜓 )
7	4 6	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑧 ∈ 𝐴 ∀ 𝑤 ∈ 𝐵 𝜓 )