Theorem ralrimdvv 3185

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 408	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜒))
3	2	ralrimivv 3182	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)
4	3	ex 414	1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2121  ∀wral 3055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918

This theorem depends on definitions:  df-bi 209  df-an 398  df-ral 3056

This theorem is referenced by:  ralrimdvva  3196  lspsneu  21119  pmatcoe1fsupp  22687  aalioulem4  26322  fargshiftf1  47928