Theorem 4ralimi

Description: Inference quantifying both antecedent and consequent four times, with strong hypothesis. (Contributed by Scott Fenton, 5-Mar-2025)

	Ref	Expression
Hypothesis	2ralimi.1	⊢ ( 𝜑 → 𝜓 )
Assertion	4ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜓 )

Proof

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