Theorem ralimdvv 3194

Description: Deduction doubly quantifying both antecedent and consequent. (Contributed by Scott Fenton, 2-Mar-2025.) Shorten and reduce DV conditions. (Revised by Eric Schmidt, 18-Nov-2025.)

Hypothesis

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

Assertion

Ref	Expression
ralimdvv	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem ralimdvv

Step	Hyp	Ref	Expression
1		ralimdvv.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	ralimdv 3155	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 𝜓 → ∀𝑦 ∈ 𝐵 𝜒))
3	2	ralimdv 3155	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910

This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3053

This theorem is referenced by:  ralimd4v  3196  ralimd6v  3198