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Theorem findes 4253
Description: Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. See tfindes 4220 for the transfinite version. (Contributed by Raph Levien, 9-Jul-2003.)
Hypotheses
Ref Expression
findes.1  |-  [ (/)  /  x ] ph
findes.2  |-  ( x  e.  om  ->  ( ph  ->  [ suc  x  /  x ] ph )
)
Assertion
Ref Expression
findes  |-  ( x  e.  om  ->  ph )

Proof of Theorem findes
StepHypRef Expression
1 dfsbcq 2652 . 2  |-  ( z  =  (/)  ->  ( [ z  /  x ] ph 
<->  [ (/)  /  x ] ph ) )
2 sbequ 1798 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
3 dfsbcq 2652 . 2  |-  ( z  =  suc  y  -> 
( [ z  /  x ] ph  <->  [ suc  y  /  x ] ph ) )
4 sbequ12r 1746 . 2  |-  ( z  =  x  ->  ( [ z  /  x ] ph  <->  ph ) )
5 findes.1 . 2  |-  [ (/)  /  x ] ph
6 ax-17 1529 . . . 4  |-  ( y  e.  om  ->  A. x  y  e.  om )
7 hbs1 1909 . . . . 5  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
8 hbsbc1 2662 . . . . 5  |-  ( [ suc  y  /  x ] ph  ->  A. x [ suc  y  /  x ] ph )
97, 8hbim 1605 . . . 4  |-  ( ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph )  ->  A. x ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph )
)
106, 9hbim 1605 . . 3  |-  ( ( y  e.  om  ->  ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph ) )  ->  A. x
( y  e.  om  ->  ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph ) ) )
11 eleq1 2124 . . . 4  |-  ( x  =  y  ->  (
x  e.  om  <->  y  e.  om ) )
12 sbequ12 1745 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
13 suceq 4008 . . . . . 6  |-  ( x  =  y  ->  suc  x  =  suc  y )
14 dfsbcq 2652 . . . . . 6  |-  ( suc  x  =  suc  y  ->  ( [ suc  x  /  x ] ph  <->  [ suc  y  /  x ] ph ) )
1513, 14syl 15 . . . . 5  |-  ( x  =  y  ->  ( [ suc  x  /  x ] ph  <->  [ suc  y  /  x ] ph ) )
1612, 15imbi12d 309 . . . 4  |-  ( x  =  y  ->  (
( ph  ->  [ suc  x  /  x ] ph ) 
<->  ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph ) ) )
1711, 16imbi12d 309 . . 3  |-  ( x  =  y  ->  (
( x  e.  om  ->  ( ph  ->  [ suc  x  /  x ] ph ) )  <->  ( y  e.  om  ->  ( [
y  /  x ] ph  ->  [ suc  y  /  x ] ph )
) ) )
18 findes.2 . . 3  |-  ( x  e.  om  ->  ( ph  ->  [ suc  x  /  x ] ph )
)
1910, 17, 18chvar 1730 . 2  |-  ( y  e.  om  ->  ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph ) )
201, 2, 3, 4, 5, 19finds 4249 1  |-  ( x  e.  om  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 174    = wceq 1520    e. wcel 1522   [wsbc 1734   (/)c0 3072   suc csuc 3945   omcom 4223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1442  ax-6 1443  ax-7 1444  ax-gen 1445  ax-8 1524  ax-11 1525  ax-13 1526  ax-14 1527  ax-17 1529  ax-12o 1562  ax-10 1576  ax-9 1582  ax-4 1589  ax-16 1775  ax-ext 2046  ax-sep 3701  ax-nul 3709  ax-pr 3769  ax-un 4061
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 897  df-3an 898  df-ex 1447  df-sb 1736  df-eu 1958  df-mo 1959  df-clab 2052  df-cleq 2057  df-clel 2060  df-ne 2184  df-ral 2278  df-rex 2279  df-rab 2281  df-v 2477  df-sbc 2651  df-dif 2796  df-un 2798  df-in 2800  df-ss 2804  df-pss 2806  df-nul 3073  df-if 3182  df-pw 3243  df-sn 3261  df-pr 3262  df-tp 3263  df-op 3264  df-uni 3425  df-br 3587  df-opab 3641  df-tr 3674  df-eprel 3856  df-po 3865  df-so 3866  df-fr 3903  df-we 3905  df-ord 3946  df-on 3947  df-lim 3948  df-suc 3949  df-om 4224
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