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Theorem raluz2 12880
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 12827 . . . . . 6 (𝑛 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀𝑛))
2 3anass 1095 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀𝑛) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
31, 2bitri 274 . . . . 5 (𝑛 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
43imbi1i 349 . . . 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 335 . . . . . 6 ((𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑)) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))))
85, 7bitri 274 . . . . 5 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))))
9 bi2.04 388 . . . . 5 ((𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
108, 9bitri 274 . . . 4 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
114, 10bitri 274 . . 3 ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
1211ralbii2 3089 . 2 (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀𝑛𝜑)))
13 r19.21v 3179 . 2 (∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀𝑛𝜑)) ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
1412, 13bitri 274 1 (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wcel 2106  wral 3061   class class class wbr 5148  cfv 6543  cle 11248  cz 12557  cuz 12821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-cnex 11165  ax-resscn 11166
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-neg 11446  df-z 12558  df-uz 12822
This theorem is referenced by: (None)
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