Theorem raluz2 12939

Description: Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)

Assertion

Ref	Expression
raluz2	⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑)))

Distinct variable group:   𝑛,𝑀

Allowed substitution hint:   𝜑(𝑛)

Proof of Theorem raluz2

Step	Hyp	Ref	Expression
1		eluz2 12884	. . . . . 6 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛))
2		3anass 1095	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)))
3	1, 2	bitri 275	. . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)))
4	3	imbi1i 349	. . . 4 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) → 𝜑) ↔ ((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) → 𝜑))
5		impexp 450	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) → 𝜑)))
6		impexp 450	. . . . . . 7 ⊢ (((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)))
7	6	imbi2i 336	. . . . . 6 ⊢ ((𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) → 𝜑)) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))))
8	5, 7	bitri 275	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))))
9		bi2.04 387	. . . . 5 ⊢ ((𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))))
10	8, 9	bitri 275	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))))
11	4, 10	bitri 275	. . 3 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))))
12	11	ralbii2 3089	. 2 ⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)))
13		r19.21v 3180	. 2 ⊢ (∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)) ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑)))
14	12, 13	bitri 275	1 ⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108  ∀wral 3061   class class class wbr 5143  ‘cfv 6561   ≤ cle 11296  ℤcz 12613  ℤ_≥cuz 12878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-cnex 11211  ax-resscn 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-neg 11495  df-z 12614  df-uz 12879

This theorem is referenced by: (None)