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Theorem raluz2 12289
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 12241 . . . . . 6 (𝑛 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀𝑛))
2 3anass 1092 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀𝑛) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
31, 2bitri 278 . . . . 5 (𝑛 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
43imbi1i 353 . . . 4 ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ ((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑))
5 impexp 454 . . . . . 6 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑)))
6 impexp 454 . . . . . . 7 (((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀𝑛𝜑)))
76imbi2i 339 . . . . . 6 ((𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑)) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))))
85, 7bitri 278 . . . . 5 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))))
9 bi2.04 392 . . . . 5 ((𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
108, 9bitri 278 . . . 4 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
114, 10bitri 278 . . 3 ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
1211ralbii2 3134 . 2 (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀𝑛𝜑)))
13 r19.21v 3145 . 2 (∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀𝑛𝜑)) ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
1412, 13bitri 278 1 (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wcel 2112  wral 3109   class class class wbr 5033  cfv 6328  cle 10669  cz 11973  cuz 12235
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-cnex 10586  ax-resscn 10587
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-neg 10866  df-z 11974  df-uz 12236
This theorem is referenced by: (None)
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