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Theorem ralimdvva 2958
Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1735). (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 679 . . 3 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32ralimdva 2956 . 2 ((𝜑𝑥𝐴) → (∀𝑦𝐵 𝜓 → ∀𝑦𝐵 𝜒))
43ralimdva 2956 1 (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836
This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2912
This theorem is referenced by:  dedekindle  10145  islmhm2  18957  dmatscmcl  20228  cpmatacl  20440  cpmatinvcl  20441  mat2pmatf1  20453  pmatcollpw2lem  20501  tgpt0  21832  isngp4  22326  addcnlem  22575  c1lip3  23666  aalioulem2  23992  aalioulem5  23995  aalioulem6  23996  aaliou  23997  iscgrglt  25309  frcond3  26998  2pthfrgrrn  27010  2pthfrgrrn2  27011  equivbnd  33221  ghomco  33322
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