MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexuz Unicode version

Theorem rexuz 10483
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 10448 . . . 4  |-  ( M  e.  ZZ  ->  (
n  e.  ( ZZ>= `  M )  <->  ( n  e.  ZZ  /\  M  <_  n ) ) )
21anbi1d 686 . . 3  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  /\  ph )  <->  ( ( n  e.  ZZ  /\  M  <_  n )  /\  ph ) ) )
3 anass 631 . . 3  |-  ( ( ( n  e.  ZZ  /\  M  <_  n )  /\  ph )  <->  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) )
42, 3syl6bb 253 . 2  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  /\  ph )  <->  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
54rexbidv2 2689 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 177    /\ wa 359    e. wcel 1721   E.wrex 2667   class class class wbr 4172   ` cfv 5413    <_ cle 9077   ZZcz 10238   ZZ>=cuz 10444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-cnex 9002  ax-resscn 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-neg 9250  df-z 10239  df-uz 10445
  Copyright terms: Public domain W3C validator