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Theorem rexuz 11698
 Description: Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
Assertion
Ref Expression
rexuz (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))
Distinct variable group:   𝑛,𝑀
Allowed substitution hint:   𝜑(𝑛)

Proof of Theorem rexuz
StepHypRef Expression
1 eluz1 11651 . . . 4 (𝑀 ∈ ℤ → (𝑛 ∈ (ℤ𝑀) ↔ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
21anbi1d 740 . . 3 (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) ↔ ((𝑛 ∈ ℤ ∧ 𝑀𝑛) ∧ 𝜑)))
3 anass 680 . . 3 (((𝑛 ∈ ℤ ∧ 𝑀𝑛) ∧ 𝜑) ↔ (𝑛 ∈ ℤ ∧ (𝑀𝑛𝜑)))
42, 3syl6bb 276 . 2 (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) ↔ (𝑛 ∈ ℤ ∧ (𝑀𝑛𝜑))))
54rexbidv2 3043 1 (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∈ wcel 1987  ∃wrex 2909   class class class wbr 4623  ‘cfv 5857   ≤ cle 10035  ℤcz 11337  ℤ≥cuz 11647 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-cnex 9952  ax-resscn 9953 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-neg 10229  df-z 11338  df-uz 11648 This theorem is referenced by: (None)
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