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Theorem raluz 8747
Description: Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
Assertion
Ref Expression
raluz  |-  ( M  e.  ZZ  ->  ( A. n  e.  ( ZZ>=
`  M ) ph  <->  A. n  e.  ZZ  ( M  <_  n  ->  ph )
) )
Distinct variable group:    n, M
Allowed substitution hint:    ph( n)

Proof of Theorem raluz
StepHypRef Expression
1 eluz1 8704 . . . 4  |-  ( M  e.  ZZ  ->  (
n  e.  ( ZZ>= `  M )  <->  ( n  e.  ZZ  /\  M  <_  n ) ) )
21imbi1d 229 . . 3  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  ->  ph )  <->  ( ( n  e.  ZZ  /\  M  <_  n )  ->  ph )
) )
3 impexp 259 . . 3  |-  ( ( ( n  e.  ZZ  /\  M  <_  n )  ->  ph )  <->  ( n  e.  ZZ  ->  ( M  <_  n  ->  ph ) ) )
42, 3syl6bb 194 . 2  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  ->  ph )  <->  ( n  e.  ZZ  ->  ( M  <_  n  ->  ph ) ) ) )
54ralbidv2 2371 1  |-  ( M  e.  ZZ  ->  ( A. n  e.  ( ZZ>=
`  M ) ph  <->  A. n  e.  ZZ  ( M  <_  n  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1434   A.wral 2349   class class class wbr 3793   ` cfv 4932    <_ cle 7216   ZZcz 8432   ZZ>=cuz 8700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-cnex 7129  ax-resscn 7130
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-ov 5546  df-neg 7349  df-z 8433  df-uz 8701
This theorem is referenced by: (None)
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