Theorem raluz2 9700

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 9654	. . . . . 6 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛))
2		3anass 985	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)))
3	1, 2	bitri 184	. . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)))
4	3	imbi1i 238	. . . 4 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) → 𝜑) ↔ ((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) → 𝜑))
5		impexp 263	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) → 𝜑)))
6		impexp 263	. . . . . . 7 ⊢ (((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)))
7	6	imbi2i 226	. . . . . 6 ⊢ ((𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) → 𝜑)) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))))
8	5, 7	bitri 184	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))))
9		bi2.04 248	. . . . 5 ⊢ ((𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))))
10	8, 9	bitri 184	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))))
11	4, 10	bitri 184	. . 3 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))))
12	11	ralbii2 2516	. 2 ⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)))
13		r19.21v 2583	. 2 ⊢ (∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)) ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑)))
14	12, 13	bitri 184	1 ⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   ∈ wcel 2176  ∀wral 2484   class class class wbr 4044  ‘cfv 5271   ≤ cle 8108  ℤcz 9372  ℤ_≥cuz 9648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-cnex 8016  ax-resscn 8017

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5947  df-neg 8246  df-z 9373  df-uz 9649

This theorem is referenced by: (None)