Theorem 4ralimi 3125

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

Hypothesis

Ref	Expression
2ralimi.1	⊢ (𝜑 → 𝜓)

Assertion

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

Proof of Theorem 4ralimi

Step	Hyp	Ref	Expression
1		2ralimi.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	ralimi 3083	. 2 ⊢ (∀𝑤 ∈ 𝐷 𝜑 → ∀𝑤 ∈ 𝐷 𝜓)
3	2	3ralimi 3124	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜓)

Syntax hints:   → wi 4  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 206  df-ral 3062

This theorem is referenced by:  5ralimi  3126