Theorem ralimdvvOLD 3189

Description: Obsolete version of ralimdvv 3188 as of 18-Nov-2025. (Contributed by Scott Fenton, 2-Mar-2025.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
ralimdvvOLD	⊢ (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 ralimdvvOLD

Step	Hyp	Ref	Expression
1		ralimdvvOLD.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	adantr 481	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒))
3	2	ralimdvva 3186	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  ∀wral 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917

This theorem depends on definitions:  df-bi 208  df-an 397  df-ral 3054

This theorem is referenced by:  ralimd4vOLD  3191  ralimd6vOLD  3193