MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fz1sbc Unicode version

Theorem fz1sbc 10875
Description: Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)
Assertion
Ref Expression
fz1sbc  |-  ( N  e.  ZZ  ->  ( A. k  e.  ( N ... N ) ph  <->  [. N  /  k ]. ph ) )
Distinct variable group:    k, N
Allowed substitution hint:    ph( k)

Proof of Theorem fz1sbc
StepHypRef Expression
1 sbc6g 3029 . 2  |-  ( N  e.  ZZ  ->  ( [. N  /  k ]. ph  <->  A. k ( k  =  N  ->  ph )
) )
2 df-ral 2561 . . 3  |-  ( A. k  e.  ( N ... N ) ph  <->  A. k
( k  e.  ( N ... N )  ->  ph ) )
3 elfz1eq 10823 . . . . . 6  |-  ( k  e.  ( N ... N )  ->  k  =  N )
4 elfz3 10822 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  ( N ... N
) )
5 eleq1 2356 . . . . . . 7  |-  ( k  =  N  ->  (
k  e.  ( N ... N )  <->  N  e.  ( N ... N ) ) )
64, 5syl5ibrcom 213 . . . . . 6  |-  ( N  e.  ZZ  ->  (
k  =  N  -> 
k  e.  ( N ... N ) ) )
73, 6impbid2 195 . . . . 5  |-  ( N  e.  ZZ  ->  (
k  e.  ( N ... N )  <->  k  =  N ) )
87imbi1d 308 . . . 4  |-  ( N  e.  ZZ  ->  (
( k  e.  ( N ... N )  ->  ph )  <->  ( k  =  N  ->  ph )
) )
98albidv 1615 . . 3  |-  ( N  e.  ZZ  ->  ( A. k ( k  e.  ( N ... N
)  ->  ph )  <->  A. k
( k  =  N  ->  ph ) ) )
102, 9syl5rbb 249 . 2  |-  ( N  e.  ZZ  ->  ( A. k ( k  =  N  ->  ph )  <->  A. k  e.  ( N ... N
) ph ) )
111, 10bitr2d 245 1  |-  ( N  e.  ZZ  ->  ( A. k  e.  ( N ... N ) ph  <->  [. N  /  k ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530    = wceq 1632    e. wcel 1696   A.wral 2556   [.wsbc 3004  (class class class)co 5874   ZZcz 10040   ...cfz 10798
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
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-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-neg 9056  df-z 10041  df-uz 10247  df-fz 10799
  Copyright terms: Public domain W3C validator