Theorem 3ralimi

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

	Ref	Expression
Hypothesis	2ralimi.1	⊢ ( 𝜑 → 𝜓 )
Assertion	3ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜓 )

Proof

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