Theorem 6ralbidv

Description: Formula-building rule for restricted universal quantifiers (deduction form.) (Contributed by Scott Fenton, 5-Mar-2025)

	Ref	Expression
Hypothesis	6ralbidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	6ralbidv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 ∀ 𝑡 ∈ 𝐸 ∀ 𝑢 ∈ 𝐹 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 ∀ 𝑡 ∈ 𝐸 ∀ 𝑢 ∈ 𝐹 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		6ralbidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	2ralbidv	⊢ ( 𝜑 → ( ∀ 𝑡 ∈ 𝐸 ∀ 𝑢 ∈ 𝐹 𝜓 ↔ ∀ 𝑡 ∈ 𝐸 ∀ 𝑢 ∈ 𝐹 𝜒 ) )
3	2	4ralbidv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 ∀ 𝑡 ∈ 𝐸 ∀ 𝑢 ∈ 𝐹 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 ∀ 𝑡 ∈ 𝐸 ∀ 𝑢 ∈ 𝐹 𝜒 ) )