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Theorem ralrimdvva 3155
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)
Hypothesis
Ref Expression
ralrimdvva.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
ralrimdvva (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 ralrimdvva
StepHypRef Expression
1 ralrimdvva.1 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
21ex 402 . . 3 (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝜓𝜒)))
32com23 86 . 2 (𝜑 → (𝜓 → ((𝑥𝐴𝑦𝐵) → 𝜒)))
43ralrimdvv 3154 1 (𝜑 → (𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  wral 3089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006
This theorem depends on definitions:  df-bi 199  df-an 386  df-ral 3094
This theorem is referenced by:  isosolem  6825  kgencn2  21689  fbunfip  22001  reconn  22959  c1lip1  24101  cdj3i  29825  poimirlem29  33927  ispridl2  34324  ispridlc  34356
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