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Theorem fz1sbc 11648
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 3314 . 2  |-  ( N  e.  ZZ  ->  ( [. N  /  k ]. ph  <->  A. k ( k  =  N  ->  ph )
) )
2 df-ral 2801 . . 3  |-  ( A. k  e.  ( N ... N ) ph  <->  A. k
( k  e.  ( N ... N )  ->  ph ) )
3 elfz1eq 11574 . . . . . 6  |-  ( k  e.  ( N ... N )  ->  k  =  N )
4 elfz3 11573 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  ( N ... N
) )
5 eleq1 2524 . . . . . . 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 317 . . . 4  |-  ( N  e.  ZZ  ->  (
( k  e.  ( N ... N )  ->  ph )  <->  ( k  =  N  ->  ph )
) )
98albidv 1680 . . 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 1368    = wceq 1370    e. wcel 1758   A.wral 2796   [.wsbc 3288  (class class class)co 6195   ZZcz 10752   ...cfz 11549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-pre-lttri 9462  ax-pre-lttrn 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-neg 9704  df-z 10753  df-uz 10968  df-fz 11550
This theorem is referenced by: (None)
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