Theorem ralimd4v

Description: Deduction quadrupally quantifying both antecedent and consequent. (Contributed by Scott Fenton, 2-Mar-2025) Reduce DV conditions. (Revised by Eric Schmidt, 18-Nov-2025)

	Ref	Expression
Hypothesis	ralimd4v.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
Assertion	ralimd4v	⊢ ( 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		ralimd4v.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2	1	ralimdvv	⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜓 → ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜒 ) )
3	2	ralimdvv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜓 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜒 ) )