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Theorem raluz2 12285
Description: Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
Assertion
Ref Expression
raluz2 (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
Distinct variable group:   𝑛,𝑀
Allowed substitution hint:   𝜑(𝑛)

Proof of Theorem raluz2
StepHypRef Expression
1 eluz2 12237 . . . . . 6 (𝑛 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀𝑛))
2 3anass 1087 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀𝑛) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
31, 2bitri 276 . . . . 5 (𝑛 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
43imbi1i 351 . . . 4 ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ ((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑))
5 impexp 451 . . . . . 6 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑)))
6 impexp 451 . . . . . . 7 (((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀𝑛𝜑)))
76imbi2i 337 . . . . . 6 ((𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑)) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))))
85, 7bitri 276 . . . . 5 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))))
9 bi2.04 389 . . . . 5 ((𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
108, 9bitri 276 . . . 4 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
114, 10bitri 276 . . 3 ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
1211ralbii2 3160 . 2 (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀𝑛𝜑)))
13 r19.21v 3172 . 2 (∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀𝑛𝜑)) ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
1412, 13bitri 276 1 (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079  wcel 2105  wral 3135   class class class wbr 5057  cfv 6348  cle 10664  cz 11969  cuz 12231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-cnex 10581  ax-resscn 10582
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-neg 10861  df-z 11970  df-uz 12232
This theorem is referenced by: (None)
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