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Theorem ralrimdvv 3111
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 → ((𝑥𝐴𝑦𝐵) → 𝜒)))
Assertion
Ref Expression
ralrimdvv (𝜑 → (𝜓 → ∀𝑥𝐴𝑦𝐵 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
StepHypRef Expression
1 ralrimdvv.1 . . . 4 (𝜑 → (𝜓 → ((𝑥𝐴𝑦𝐵) → 𝜒)))
21imp 444 . . 3 ((𝜑𝜓) → ((𝑥𝐴𝑦𝐵) → 𝜒))
32ralrimivv 3108 . 2 ((𝜑𝜓) → ∀𝑥𝐴𝑦𝐵 𝜒)
43ex 449 1 (𝜑 → (𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  wral 3050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988
This theorem depends on definitions:  df-bi 197  df-an 385  df-ral 3055
This theorem is referenced by:  ralrimdvva  3112  lspsneu  19345  pmatcoe1fsupp  20728  aalioulem4  24309  fargshiftf1  41905
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