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Theorem rexuz 9270
Description: Restricted existential quantification in an upper set of 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 9225 . . . 4  |-  ( M  e.  ZZ  ->  (
n  e.  ( ZZ>= `  M )  <->  ( n  e.  ZZ  /\  M  <_  n ) ) )
21anbi1d 458 . . 3  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  /\  ph )  <->  ( ( n  e.  ZZ  /\  M  <_  n )  /\  ph ) ) )
3 anass 396 . . 3  |-  ( ( ( n  e.  ZZ  /\  M  <_  n )  /\  ph )  <->  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) )
42, 3syl6bb 195 . 2  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  /\  ph )  <->  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
54rexbidv2 2412 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    /\ wa 103    <-> wb 104    e. wcel 1461   E.wrex 2389   class class class wbr 3893   ` cfv 5079    <_ cle 7718   ZZcz 8951   ZZ>=cuz 9221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-cnex 7629  ax-resscn 7630
This theorem depends on definitions:  df-bi 116  df-3or 944  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-ov 5729  df-neg 7852  df-z 8952  df-uz 9222
This theorem is referenced by: (None)
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