Theorem rexuz 9819

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

Assertion

Ref	Expression
rexuz	⊢ (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑)))

Distinct variable group:   𝑛,𝑀

Allowed substitution hint:   𝜑(𝑛)

Proof of Theorem rexuz

Step	Hyp	Ref	Expression
1		eluz1 9764	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑛 ∈ (ℤ≥‘𝑀) ↔ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)))
2	1	anbi1d 465	. . 3 ⊢ (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝜑) ↔ ((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) ∧ 𝜑)))
3		anass 401	. . 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	rexbidv2 2534	1 ⊢ (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2201  ∃wrex 2510   class class class wbr 4089  ‘cfv 5328   ≤ cle 8220  ℤcz 9484  ℤ_≥cuz 9760

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 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-cnex 8128  ax-resscn 8129

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-sbc 3031  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-iota 5288  df-fun 5330  df-fv 5336  df-ov 6026  df-neg 8358  df-z 9485  df-uz 9761

This theorem is referenced by: (None)