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Theorem rexuz 8619
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 8573 . . . 4  |-  ( M  e.  ZZ  ->  (
n  e.  ( ZZ>= `  M )  <->  ( n  e.  ZZ  /\  M  <_  n ) ) )
21anbi1d 446 . . 3  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  /\  ph )  <->  ( ( n  e.  ZZ  /\  M  <_  n )  /\  ph ) ) )
3 anass 387 . . 3  |-  ( ( ( n  e.  ZZ  /\  M  <_  n )  /\  ph )  <->  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) )
42, 3syl6bb 189 . 2  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  /\  ph )  <->  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
54rexbidv2 2346 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 101    <-> wb 102    e. wcel 1409   E.wrex 2324   class class class wbr 3792   ` cfv 4930    <_ cle 7120   ZZcz 8302   ZZ>=cuz 8569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-cnex 7033  ax-resscn 7034
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-neg 7248  df-z 8303  df-uz 8570
This theorem is referenced by: (None)
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