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Theorem raluz 12910
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 12856 . . . 4 (𝑀 ∈ ℤ → (𝑛 ∈ (ℤ𝑀) ↔ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
21imbi1d 341 . . 3 (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ ((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑)))
3 impexp 450 . . 3 (((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀𝑛𝜑)))
42, 3bitrdi 287 . 2 (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀𝑛𝜑))))
54ralbidv2 3170 1 (𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2099  wral 3058   class class class wbr 5148  cfv 6548  cle 11279  cz 12588  cuz 12852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-cnex 11194  ax-resscn 11195
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-neg 11477  df-z 12589  df-uz 12853
This theorem is referenced by: (None)
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