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Theorem rexuz 10361
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 10326 . . . 4  |-  ( M  e.  ZZ  ->  (
n  e.  ( ZZ>= `  M )  <->  ( n  e.  ZZ  /\  M  <_  n ) ) )
21anbi1d 685 . . 3  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  /\  ph )  <->  ( ( n  e.  ZZ  /\  M  <_  n )  /\  ph ) ) )
3 anass 630 . . 3  |-  ( ( ( n  e.  ZZ  /\  M  <_  n )  /\  ph )  <->  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) )
42, 3syl6bb 252 . 2  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  /\  ph )  <->  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
54rexbidv2 2642 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 176    /\ wa 358    e. wcel 1710   E.wrex 2620   class class class wbr 4104   ` cfv 5337    <_ cle 8958   ZZcz 10116   ZZ>=cuz 10322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-cnex 8883  ax-resscn 8884
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-neg 9130  df-z 10117  df-uz 10323
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