Theorem raluz2 12937

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 12882	. . . . . 6 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛))
2		3anass 1094	. . . . . 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 3087	. 2 ⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)))
13		r19.21v 3178	. 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 1086   ∈ wcel 2106  ∀wral 3059   class class class wbr 5148  ‘cfv 6563   ≤ cle 11294  ℤcz 12611  ℤ_≥cuz 12876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-cnex 11209  ax-resscn 11210

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-neg 11493  df-z 12612  df-uz 12877

This theorem is referenced by: (None)