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Theorem rexuz 10532
Description: Restricted existential quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)
Assertion
Ref Expression
rexuz  |-  ( M  e.  ZZ  ->  ( E. n  e.  ( ZZ>=
`  M ) ph  <->  E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
Distinct variable group:    n, M
Allowed substitution hint:    ph( n)

Proof of Theorem rexuz
StepHypRef Expression
1 eluz1 10497 . . . 4  |-  ( M  e.  ZZ  ->  (
n  e.  ( ZZ>= `  M )  <->  ( n  e.  ZZ  /\  M  <_  n ) ) )
21anbi1d 687 . . 3  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  /\  ph )  <->  ( ( n  e.  ZZ  /\  M  <_  n )  /\  ph ) ) )
3 anass 632 . . 3  |-  ( ( ( n  e.  ZZ  /\  M  <_  n )  /\  ph )  <->  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) )
42, 3syl6bb 254 . 2  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  /\  ph )  <->  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
54rexbidv2 2730 1  |-  ( M  e.  ZZ  ->  ( E. n  e.  ( ZZ>=
`  M ) ph  <->  E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726   E.wrex 2708   class class class wbr 4215   ` cfv 5457    <_ cle 9126   ZZcz 10287   ZZ>=cuz 10493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-cnex 9051  ax-resscn 9052
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-neg 9299  df-z 10288  df-uz 10494
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