Theorem 4ralbidv

Description: Formula-building rule for restricted universal quantifiers (deduction form.) (Contributed by Scott Fenton, 20-Feb-2025)

	Ref	Expression
Hypothesis	4ralbidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	4ralbidv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		4ralbidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	ralbidv	⊢ ( 𝜑 → ( ∀ 𝑤 ∈ 𝐷 𝜓 ↔ ∀ 𝑤 ∈ 𝐷 𝜒 ) )
3	2	3ralbidv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜒 ) )