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Theorem rexuz2 10270
Description: Restricted existential quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)
Assertion
Ref Expression
rexuz2  |-  ( E. n  e.  ( ZZ>= `  M ) ph  <->  ( M  e.  ZZ  /\  E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
Distinct variable group:    n, M
Allowed substitution hint:    ph( n)

Proof of Theorem rexuz2
StepHypRef Expression
1 eluz2 10236 . . . . . 6  |-  ( n  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  n  e.  ZZ  /\  M  <_  n ) )
2 df-3an 936 . . . . . 6  |-  ( ( M  e.  ZZ  /\  n  e.  ZZ  /\  M  <_  n )  <->  ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n ) )
31, 2bitri 240 . . . . 5  |-  ( n  e.  ( ZZ>= `  M
)  <->  ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n ) )
43anbi1i 676 . . . 4  |-  ( ( n  e.  ( ZZ>= `  M )  /\  ph ) 
<->  ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n )  /\  ph ) )
5 anass 630 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n )  /\  ph ) 
<->  ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  ( M  <_  n  /\  ph ) ) )
6 anass 630 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  ( M  <_  n  /\  ph ) )  <-> 
( M  e.  ZZ  /\  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
7 an12 772 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) )  <-> 
( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
86, 7bitri 240 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  ( M  <_  n  /\  ph ) )  <-> 
( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
95, 8bitri 240 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n )  /\  ph ) 
<->  ( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
104, 9bitri 240 . . 3  |-  ( ( n  e.  ( ZZ>= `  M )  /\  ph ) 
<->  ( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
1110rexbii2 2572 . 2  |-  ( E. n  e.  ( ZZ>= `  M ) ph  <->  E. n  e.  ZZ  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) )
12 r19.42v 2694 . 2  |-  ( E. n  e.  ZZ  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) )  <->  ( M  e.  ZZ  /\  E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
1311, 12bitri 240 1  |-  ( E. n  e.  ( ZZ>= `  M ) ph  <->  ( M  e.  ZZ  /\  E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255    <_ cle 8868   ZZcz 10024   ZZ>=cuz 10230
This theorem is referenced by:  2rexuz  10271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-neg 9040  df-z 10025  df-uz 10231
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