Theorem ralrimdvv 3189

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)

Hypothesis

Ref	Expression
ralrimdvv.1	⊢ (𝜑 → (𝜓 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜒)))

Assertion

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

Distinct variable groups:   𝑥,𝑦,𝜑   𝜓,𝑥,𝑦   𝑦,𝐴

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

Proof of Theorem ralrimdvv

Step	Hyp	Ref	Expression
1		ralrimdvv.1	. . . 4 ⊢ (𝜑 → (𝜓 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜒)))
2	1	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜒))
3	2	ralrimivv 3186	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)
4	3	ex 412	1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀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:  ralrimdvva  3200  lspsneu  21089  pmatcoe1fsupp  22644  aalioulem4  26300  fargshiftf1  47422