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Theorem fz1sbc 9060
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 2811 . 2  |-  ( N  e.  ZZ  ->  ( [. N  /  k ]. ph  <->  A. k ( k  =  N  ->  ph )
) )
2 df-ral 2328 . . 3  |-  ( A. k  e.  ( N ... N ) ph  <->  A. k
( k  e.  ( N ... N )  ->  ph ) )
3 elfz1eq 9001 . . . . . 6  |-  ( k  e.  ( N ... N )  ->  k  =  N )
4 elfz3 9000 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  ( N ... N
) )
5 eleq1 2116 . . . . . . 7  |-  ( k  =  N  ->  (
k  e.  ( N ... N )  <->  N  e.  ( N ... N ) ) )
64, 5syl5ibrcom 150 . . . . . 6  |-  ( N  e.  ZZ  ->  (
k  =  N  -> 
k  e.  ( N ... N ) ) )
73, 6impbid2 135 . . . . 5  |-  ( N  e.  ZZ  ->  (
k  e.  ( N ... N )  <->  k  =  N ) )
87imbi1d 224 . . . 4  |-  ( N  e.  ZZ  ->  (
( k  e.  ( N ... N )  ->  ph )  <->  ( k  =  N  ->  ph )
) )
98albidv 1721 . . 3  |-  ( N  e.  ZZ  ->  ( A. k ( k  e.  ( N ... N
)  ->  ph )  <->  A. k
( k  =  N  ->  ph ) ) )
102, 9syl5rbb 186 . 2  |-  ( N  e.  ZZ  ->  ( A. k ( k  =  N  ->  ph )  <->  A. k  e.  ( N ... N
) ph ) )
111, 10bitr2d 182 1  |-  ( N  e.  ZZ  ->  ( A. k  e.  ( N ... N ) ph  <->  [. N  /  k ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257    = wceq 1259    e. wcel 1409   A.wral 2323   [.wsbc 2787  (class class class)co 5540   ZZcz 8302   ...cfz 8976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-pre-ltirr 7054  ax-pre-apti 7057
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-neg 7248  df-z 8303  df-uz 8570  df-fz 8977
This theorem is referenced by: (None)
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