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

Theorem ralrimd 3138
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)
Hypotheses
Ref Expression
ralrimd.1 𝑥𝜑
ralrimd.2 𝑥𝜓
ralrimd.3 (𝜑 → (𝜓 → (𝑥𝐴𝜒)))
Assertion
Ref Expression
ralrimd (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))

Proof of Theorem ralrimd
StepHypRef Expression
1 ralrimd.1 . . 3 𝑥𝜑
2 ralrimd.2 . . 3 𝑥𝜓
3 ralrimd.3 . . 3 (𝜑 → (𝜓 → (𝑥𝐴𝜒)))
41, 2, 3alrimd 2250 . 2 (𝜑 → (𝜓 → ∀𝑥(𝑥𝐴𝜒)))
5 df-ral 3092 . 2 (∀𝑥𝐴 𝜒 ↔ ∀𝑥(𝑥𝐴𝜒))
64, 5syl6ibr 244 1 (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1651  wnf 1879  wcel 2157  wral 3087
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  ax-6 2072  ax-7 2107  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-ex 1876  df-nf 1880  df-ral 3092
This theorem is referenced by:  reusv2lem3  5068  fliftfun  6788  mapxpen  8366  domtriomlem  9550  dedekind  10488  fzrevral  12675  matunitlindflem2  33886  riotasv3d  34972  ssralv2  39504  setrec1lem2  43221
  Copyright terms: Public domain W3C validator