Theorem ralimd6v 3216

Description: Deduction sextupally quantifying both antecedent and consequent. (Contributed by Scott Fenton, 5-Mar-2025.) Reduce DV conditions. (Revised by Eric Schmidt, 18-Nov-2025.)

Hypothesis

Ref	Expression
ralim6dv.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
ralimd6v	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜒))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦   𝜑,𝑧   𝜑,𝑤   𝜑,𝑝   𝜑,𝑞

Allowed substitution hints:   𝜓(𝑥, 𝑦, 𝑧, 𝑤, 𝑞, 𝑝)   𝜒(𝑥, 𝑦, 𝑧, 𝑤, 𝑞, 𝑝)   𝐴(𝑥, 𝑦, 𝑧, 𝑤, 𝑞, 𝑝)   𝐵(𝑥, 𝑦, 𝑧, 𝑤, 𝑞, 𝑝)   𝐶(𝑥, 𝑦, 𝑧, 𝑤, 𝑞, 𝑝)   𝐷(𝑥, 𝑦, 𝑧, 𝑤, 𝑞, 𝑝)   𝐸(𝑥, 𝑦, 𝑧, 𝑤, 𝑞, 𝑝)   𝐹(𝑥, 𝑦, 𝑧, 𝑤, 𝑞, 𝑝)

Proof of Theorem ralimd6v

Step	Hyp	Ref	Expression
1		ralim6dv.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	ralimdvv 3212	. 2 ⊢ (𝜑 → (∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜓 → ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜒))
3	2	ralimd4v 3214	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜒))

This proof depends on syntax axioms:   → wi 4  ∀wral 3077

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-ral 3078

This theorem is used by:  mulsproplem13  28514  mulsproplem14  28515