Theorem ralimd4v 3207

Description: Deduction quadrupally quantifying both antecedent and consequent. (Contributed by Scott Fenton, 2-Mar-2025.)

Hypothesis

Ref	Expression
ralimd4v.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
ralimd4v	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜒))

Distinct variable groups:   𝑦,𝑧,𝑤,𝐴   𝑧,𝐵,𝑤   𝑤,𝐶   𝑥,𝑦,𝑧,𝑤,𝜑

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

Proof of Theorem ralimd4v

Step	Hyp	Ref	Expression
1		ralimd4v.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	ralimdvv 3206	. 2 ⊢ (𝜑 → (∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜓 → ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜒))
3	2	ralimdvv 3206	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜒))

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  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-an 397  df-ral 3062

This theorem is referenced by:  ralimd6v  3208