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Theorem rexuz2 10529
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 10495 . . . . . 6  |-  ( n  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  n  e.  ZZ  /\  M  <_  n ) )
2 df-3an 939 . . . . . 6  |-  ( ( M  e.  ZZ  /\  n  e.  ZZ  /\  M  <_  n )  <->  ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n ) )
31, 2bitri 242 . . . . 5  |-  ( n  e.  ( ZZ>= `  M
)  <->  ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n ) )
43anbi1i 678 . . . 4  |-  ( ( n  e.  ( ZZ>= `  M )  /\  ph ) 
<->  ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n )  /\  ph ) )
5 anass 632 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n )  /\  ph ) 
<->  ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  ( M  <_  n  /\  ph ) ) )
6 anass 632 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  ( M  <_  n  /\  ph ) )  <-> 
( M  e.  ZZ  /\  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
7 an12 774 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) )  <-> 
( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
86, 7bitri 242 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  ( M  <_  n  /\  ph ) )  <-> 
( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
95, 8bitri 242 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n )  /\  ph ) 
<->  ( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
104, 9bitri 242 . . 3  |-  ( ( n  e.  ( ZZ>= `  M )  /\  ph ) 
<->  ( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
1110rexbii2 2735 . 2  |-  ( E. n  e.  ( ZZ>= `  M ) ph  <->  E. n  e.  ZZ  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) )
12 r19.42v 2863 . 2  |-  ( E. n  e.  ZZ  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) )  <->  ( M  e.  ZZ  /\  E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
1311, 12bitri 242 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 178    /\ wa 360    /\ w3a 937    e. wcel 1726   E.wrex 2707   class class class wbr 4213   ` cfv 5455    <_ cle 9122   ZZcz 10283   ZZ>=cuz 10489
This theorem is referenced by:  2rexuz  10530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-cnex 9047  ax-resscn 9048
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-neg 9295  df-z 10284  df-uz 10490
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