Theorem ralimd6v 3209

Description: Deduction sextupally quantifying both antecedent and consequent. (Contributed by Scott Fenton, 5-Mar-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:   𝑦,𝑧,𝑤,𝐴   𝑧,𝐵,𝑤   𝑤,𝐶   𝐸,𝑞   𝑥,𝑦,𝑧,𝑤,𝜑   𝑞,𝑝,𝜑

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑤,𝑞,𝑝)   𝜒(𝑥,𝑦,𝑧,𝑤,𝑞,𝑝)   𝐴(𝑥,𝑞,𝑝)   𝐵(𝑥,𝑦,𝑞,𝑝)   𝐶(𝑥,𝑦,𝑧,𝑞,𝑝)   𝐷(𝑥,𝑦,𝑧,𝑤,𝑞,𝑝)   𝐸(𝑥,𝑦,𝑧,𝑤,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 3207	. 2 ⊢ (𝜑 → (∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜓 → ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜒))
3	2	ralimd4v 3208	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))

Syntax hints:   → wi 4  ∀wral 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909

This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3061

This theorem is referenced by:  mulsproplem13  28155  mulsproplem14  28156