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

Theorem ralimdvva 2947
 Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1729). (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 678 . . 3 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32ralimdva 2945 . 2 ((𝜑𝑥𝐴) → (∀𝑦𝐵 𝜓 → ∀𝑦𝐵 𝜒))
43ralimdva 2945 1 (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827 This theorem depends on definitions:  df-bi 196  df-an 385  df-ral 2901 This theorem is referenced by:  dedekindle  10080  islmhm2  18859  dmatscmcl  20128  cpmatacl  20340  cpmatinvcl  20341  mat2pmatf1  20353  pmatcollpw2lem  20401  tgpt0  21732  isngp4  22226  addcnlem  22475  c1lip3  23566  aalioulem2  23892  aalioulem5  23895  aalioulem6  23896  aaliou  23897  iscgrglt  25209  nofulllem4  31104  equivbnd  32759  ghomco  32860  frcond3  41440  2pthfrgrrn  41452  2pthfrgrrn2  41453
 Copyright terms: Public domain W3C validator