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Theorem findes 4297
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 4264 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 2689 . 2  |-  ( z  =  (/)  ->  ( [ z  /  x ] ph 
<->  [ (/)  /  x ] ph ) )
2 sbequ 1810 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
3 dfsbcq 2689 . 2  |-  ( z  =  suc  y  -> 
( [ z  /  x ] ph  <->  [ suc  y  /  x ] ph ) )
4 sbequ12r 1758 . 2  |-  ( z  =  x  ->  ( [ z  /  x ] ph  <->  ph ) )
5 findes.1 . 2  |-  [ (/)  /  x ] ph
6 ax-17 1540 . . . 4  |-  ( y  e.  om  ->  A. x  y  e.  om )
7 hbs1 1921 . . . . 5  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
8 hbsbc1 2699 . . . . 5  |-  ( [ suc  y  /  x ] ph  ->  A. x [ suc  y  /  x ] ph )
97, 8hbim 1617 . . . 4  |-  ( ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph )  ->  A. x ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph )
)
106, 9hbim 1617 . . 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 2160 . . . 4  |-  ( x  =  y  ->  (
x  e.  om  <->  y  e.  om ) )
12 sbequ12 1757 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
13 suceq 4052 . . . . . 6  |-  ( x  =  y  ->  suc  x  =  suc  y )
14 dfsbcq 2689 . . . . . 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 1742 . 2  |-  ( y  e.  om  ->  ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph ) )
201, 2, 3, 4, 5, 19finds 4293 1  |-  ( x  e.  om  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 174    = wceq 1531    e. wcel 1533   [wsbc 1746   (/)c0 3110   suc csuc 3989   omcom 4267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1452  ax-6 1453  ax-7 1454  ax-gen 1455  ax-8 1535  ax-11 1536  ax-13 1537  ax-14 1538  ax-17 1540  ax-12o 1574  ax-10 1588  ax-9 1594  ax-4 1601  ax-16 1787  ax-ext 2082  ax-sep 3745  ax-nul 3753  ax-pr 3813  ax-un 4105
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 901  df-3an 902  df-ex 1457  df-sb 1748  df-eu 1970  df-mo 1971  df-clab 2088  df-cleq 2093  df-clel 2096  df-ne 2220  df-ral 2315  df-rex 2316  df-rab 2318  df-v 2514  df-sbc 2688  df-dif 2833  df-un 2835  df-in 2837  df-ss 2841  df-pss 2843  df-nul 3111  df-if 3221  df-pw 3282  df-sn 3300  df-pr 3301  df-tp 3302  df-op 3303  df-uni 3469  df-br 3631  df-opab 3685  df-tr 3718  df-eprel 3900  df-po 3909  df-so 3910  df-fr 3947  df-we 3949  df-ord 3990  df-on 3991  df-lim 3992  df-suc 3993  df-om 4268
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