Theorem ralimdvva 3179

Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1811). (Contributed by AV, 27-Nov-2019.)

Hypothesis

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

Assertion

Ref	Expression
ralimdvva	⊢ (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 ralimdvva

Step	Hyp	Ref	Expression
1		ralimdvva.1	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒))
2	1	anassrs 467	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))
3	2	ralimdva 3144	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 𝜓 → ∀𝑦 ∈ 𝐵 𝜒))
4	3	ralimdva 3144	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2111  ∀wral 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3048

This theorem is referenced by:  ralimdvvOLD  3182  dedekindle  11277  isdomn4  20631  islmhm2  20972  dflidl2rng  21155  dmatscmcl  22418  cpmatacl  22631  cpmatinvcl  22632  mat2pmatf1  22644  pmatcollpw2lem  22692  tgpt0  24034  isngp4  24527  addcnlem  24780  c1lip3  25931  aalioulem2  26268  aalioulem5  26271  aalioulem6  26272  aaliou  26273  iscgrglt  28492  2pthfrgrrn  30262  2pthfrgrrn2  30263  equivbnd  37840  ghomco  37941  fcoresf1  47179  fullthinc  49561