Theorem 5ralimi 3140

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

Hypothesis

Ref	Expression
2ralimi.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
5ralimi	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 𝜓)

Proof of Theorem 5ralimi

Step	Hyp	Ref	Expression
1		2ralimi.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	ralimi 3104	. 2 ⊢ (∀𝑡 ∈ 𝐸 𝜑 → ∀𝑡 ∈ 𝐸 𝜓)
3	2	4ralimi 3139	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wral 3081

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-ral 3082

This theorem is used by:  6ralimi  3141