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Theorem raluz 10517
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 10484 . . . 4  |-  ( M  e.  ZZ  ->  (
n  e.  ( ZZ>= `  M )  <->  ( n  e.  ZZ  /\  M  <_  n ) ) )
21imbi1d 309 . . 3  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  ->  ph )  <->  ( ( n  e.  ZZ  /\  M  <_  n )  ->  ph )
) )
3 impexp 434 . . 3  |-  ( ( ( n  e.  ZZ  /\  M  <_  n )  ->  ph )  <->  ( n  e.  ZZ  ->  ( M  <_  n  ->  ph ) ) )
42, 3syl6bb 253 . 2  |-  ( M  e.  ZZ  ->  (
( n  e.  (
ZZ>= `  M )  ->  ph )  <->  ( n  e.  ZZ  ->  ( M  <_  n  ->  ph ) ) ) )
54ralbidv2 2719 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 177    /\ wa 359    e. wcel 1725   A.wral 2697   class class class wbr 4204   ` cfv 5446    <_ cle 9113   ZZcz 10274   ZZ>=cuz 10480
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-neg 9286  df-z 10275  df-uz 10481
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