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Theorem fz1sbc 11758
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 3350 . 2  |-  ( N  e.  ZZ  ->  ( [. N  /  k ]. ph  <->  A. k ( k  =  N  ->  ph )
) )
2 df-ral 2809 . . 3  |-  ( A. k  e.  ( N ... N ) ph  <->  A. k
( k  e.  ( N ... N )  ->  ph ) )
3 elfz1eq 11700 . . . . . 6  |-  ( k  e.  ( N ... N )  ->  k  =  N )
4 elfz3 11699 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  ( N ... N
) )
5 eleq1 2526 . . . . . . 7  |-  ( k  =  N  ->  (
k  e.  ( N ... N )  <->  N  e.  ( N ... N ) ) )
64, 5syl5ibrcom 222 . . . . . 6  |-  ( N  e.  ZZ  ->  (
k  =  N  -> 
k  e.  ( N ... N ) ) )
73, 6impbid2 204 . . . . 5  |-  ( N  e.  ZZ  ->  (
k  e.  ( N ... N )  <->  k  =  N ) )
87imbi1d 315 . . . 4  |-  ( N  e.  ZZ  ->  (
( k  e.  ( N ... N )  ->  ph )  <->  ( k  =  N  ->  ph )
) )
98albidv 1718 . . 3  |-  ( N  e.  ZZ  ->  ( A. k ( k  e.  ( N ... N
)  ->  ph )  <->  A. k
( k  =  N  ->  ph ) ) )
102, 9syl5rbb 258 . 2  |-  ( N  e.  ZZ  ->  ( A. k ( k  =  N  ->  ph )  <->  A. k  e.  ( N ... N
) ph ) )
111, 10bitr2d 254 1  |-  ( N  e.  ZZ  ->  ( A. k  e.  ( N ... N ) ph  <->  [. N  /  k ]. ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1396    = wceq 1398    e. wcel 1823   A.wral 2804   [.wsbc 3324  (class class class)co 6270   ZZcz 10860   ...cfz 11675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-neg 9799  df-z 10861  df-uz 11083  df-fz 11676
This theorem is referenced by: (None)
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