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Theorem ralimdvva 3204
Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1807). (Contributed by AV, 27-Nov-2019.)
Hypothesis
Ref Expression
ralimdvva.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
ralimdvva (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 → ∀𝑥𝐴𝑦𝐵 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
StepHypRef Expression
1 ralimdvva.1 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
21anassrs 467 . . 3 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32ralimdva 3165 . 2 ((𝜑𝑥𝐴) → (∀𝑦𝐵 𝜓 → ∀𝑦𝐵 𝜒))
43ralimdva 3165 1 (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908
This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3060
This theorem is referenced by:  ralimdvv  3206  dedekindle  11423  isdomn4  20733  islmhm2  21055  dflidl2rng  21246  dmatscmcl  22525  cpmatacl  22738  cpmatinvcl  22739  mat2pmatf1  22751  pmatcollpw2lem  22799  tgpt0  24143  isngp4  24641  addcnlem  24900  c1lip3  26053  aalioulem2  26390  aalioulem5  26393  aalioulem6  26394  aaliou  26395  iscgrglt  28537  2pthfrgrrn  30311  2pthfrgrrn2  30312  equivbnd  37777  ghomco  37878  fcoresf1  47019  fullthinc  48846
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