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Theorem rexuz 12295
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 12244 . . . 4 (𝑀 ∈ ℤ → (𝑛 ∈ (ℤ𝑀) ↔ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
21anbi1d 632 . . 3 (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) ↔ ((𝑛 ∈ ℤ ∧ 𝑀𝑛) ∧ 𝜑)))
3 anass 472 . . 3 (((𝑛 ∈ ℤ ∧ 𝑀𝑛) ∧ 𝜑) ↔ (𝑛 ∈ ℤ ∧ (𝑀𝑛𝜑)))
42, 3syl6bb 290 . 2 (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) ↔ (𝑛 ∈ ℤ ∧ (𝑀𝑛𝜑))))
54rexbidv2 3287 1 (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  wrex 3134   class class class wbr 5052  cfv 6343  cle 10674  cz 11978  cuz 12240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-cnex 10591  ax-resscn 10592
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-neg 10871  df-z 11979  df-uz 12241
This theorem is referenced by: (None)
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