Theorem ralimdvv 3197

Description: Deduction doubly quantifying both antecedent and consequent. (Contributed by Scott Fenton, 2-Mar-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:   𝑦,𝐴   𝑥,𝑦,𝜑

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

Proof of Theorem ralimdvv

Step	Hyp	Ref	Expression
1		ralimdvv.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	adantr 479	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒))
3	2	ralimdvva 3195	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2099  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906

This theorem depends on definitions:  df-bi 206  df-an 395  df-ral 3052

This theorem is referenced by:  ralimd4v  3198  ralimd6v  3199