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Theorem raluz2 11681
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 11637 . . . . . 6 (𝑛 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀𝑛))
2 3anass 1040 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀𝑛) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
31, 2bitri 264 . . . . 5 (𝑛 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
43imbi1i 339 . . . 4 ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ ((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑))
5 impexp 462 . . . . . 6 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑)))
6 impexp 462 . . . . . . 7 (((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀𝑛𝜑)))
76imbi2i 326 . . . . . 6 ((𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑)) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))))
85, 7bitri 264 . . . . 5 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))))
9 bi2.04 376 . . . . 5 ((𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
108, 9bitri 264 . . . 4 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
114, 10bitri 264 . . 3 ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
1211ralbii2 2972 . 2 (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀𝑛𝜑)))
13 r19.21v 2954 . 2 (∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀𝑛𝜑)) ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
1412, 13bitri 264 1 (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wcel 1987  wral 2907   class class class wbr 4613  cfv 5847  cle 10019  cz 11321  cuz 11631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-cnex 9936  ax-resscn 9937
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-neg 10213  df-z 11322  df-uz 11632
This theorem is referenced by: (None)
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