MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralrimdvva Structured version   Visualization version   GIF version

Theorem ralrimdvva 3200
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 414 . . 3 (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝜓𝜒)))
32com23 86 . 2 (𝜑 → (𝜓 → ((𝑥𝐴𝑦𝐵) → 𝜒)))
43ralrimdvv 3195 1 (𝜑 → (𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914
This theorem depends on definitions:  df-bi 206  df-an 398  df-ral 3062
This theorem is referenced by:  isosolem  7293  kgencn2  22924  fbunfip  23236  reconn  24207  c1lip1  25377  cdj3i  31425  poimirlem29  36153  ispridl2  36543  ispridlc  36575
  Copyright terms: Public domain W3C validator