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

Theorem raluz 11612
Description: Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
Assertion
Ref Expression
raluz (𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
Distinct variable group:   𝑛,𝑀
Allowed substitution hint:   𝜑(𝑛)

Proof of Theorem raluz
StepHypRef Expression
1 eluz1 11567 . . . 4 (𝑀 ∈ ℤ → (𝑛 ∈ (ℤ𝑀) ↔ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
21imbi1d 330 . . 3 (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ ((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑)))
3 impexp 461 . . 3 (((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀𝑛𝜑)))
42, 3syl6bb 275 . 2 (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀𝑛𝜑))))
54ralbidv2 2967 1 (𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wcel 1977  wral 2896   class class class wbr 4583  cfv 5804  cle 9954  cz 11254  cuz 11563
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  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-cnex 9871  ax-resscn 9872
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-neg 10148  df-z 11255  df-uz 11564
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator