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Metamath Proof Explorer

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 )