Theorem raluz 9811

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

Assertion

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

Distinct variable group:   𝑛,𝑀

Allowed substitution hint:   𝜑(𝑛)

Proof of Theorem raluz

Step	Hyp	Ref	Expression
1		eluz1 9758	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑛 ∈ (ℤ≥‘𝑀) ↔ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)))
2	1	imbi1d 231	. . 3 ⊢ (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ≥‘𝑀) → 𝜑) ↔ ((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) → 𝜑)))
3		impexp 263	. . 3 ⊢ (((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)))
4	2, 3	bitrdi 196	. 2 ⊢ (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ≥‘𝑀) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))))
5	4	ralbidv2 2534	1 ⊢ (𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2202  ∀wral 2510   class class class wbr 4088  ‘cfv 5326   ≤ cle 8214  ℤcz 9478  ℤ_≥cuz 9754

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-cnex 8122  ax-resscn 8123

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-neg 8352  df-z 9479  df-uz 9755

This theorem is referenced by: (None)