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Theorem raluz 10283
Description: Restricted universal quantification in a set of upper 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 10250 . . . 4  |-  ( M  e.  ZZ  ->  (
n  e.  ( ZZ>= `  M )  <->  ( n  e.  ZZ  /\  M  <_  n ) ) )
21imbi1d 308 . . 3  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  ->  ph )  <->  ( ( n  e.  ZZ  /\  M  <_  n )  ->  ph )
) )
3 impexp 433 . . 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 ) ) ) )
54ralbidv2 2578 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    <-> wb 176    /\ wa 358    e. wcel 1696   A.wral 2556   class class class wbr 4039   ` cfv 5271    <_ cle 8884   ZZcz 10040   ZZ>=cuz 10246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-cnex 8809  ax-resscn 8810
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-neg 9056  df-z 10041  df-uz 10247
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