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Theorem findes 4259
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 4226 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 2653 . 2  |-  ( z  =  (/)  ->  ( [ z  /  x ] ph 
<->  [ (/)  /  x ] ph ) )
2 sbequ 1799 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
3 dfsbcq 2653 . 2  |-  ( z  =  suc  y  -> 
( [ z  /  x ] ph  <->  [ suc  y  /  x ] ph ) )
4 sbequ12r 1747 . 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 1910 . . . . 5  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
8 hbsbc1 2663 . . . . 5  |-  ( [ suc  y  /  x ] ph  ->  A. x [ suc  y  /  x ] ph )
97, 8hbim 1606 . . . 4  |-  ( ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph )  ->  A. x ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph )
)
106, 9hbim 1606 . . 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 2125 . . . 4  |-  ( x  =  y  ->  (
x  e.  om  <->  y  e.  om ) )
12 sbequ12 1746 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
13 suceq 4014 . . . . . 6  |-  ( x  =  y  ->  suc  x  =  suc  y )
14 dfsbcq 2653 . . . . . 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 1731 . 2  |-  ( y  e.  om  ->  ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph ) )
201, 2, 3, 4, 5, 19finds 4255 1  |-  ( x  e.  om  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 174    = wceq 1520    e. wcel 1522   [wsbc 1735   (/)c0 3073   suc csuc 3951   omcom 4229
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 1563  ax-10 1577  ax-9 1583  ax-4 1590  ax-16 1776  ax-ext 2047  ax-sep 3707  ax-nul 3715  ax-pr 3775  ax-un 4067
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 1737  df-eu 1959  df-mo 1960  df-clab 2053  df-cleq 2058  df-clel 2061  df-ne 2185  df-ral 2279  df-rex 2280  df-rab 2282  df-v 2478  df-sbc 2652  df-dif 2797  df-un 2799  df-in 2801  df-ss 2805  df-pss 2807  df-nul 3074  df-if 3183  df-pw 3244  df-sn 3262  df-pr 3263  df-tp 3264  df-op 3265  df-uni 3431  df-br 3593  df-opab 3647  df-tr 3680  df-eprel 3862  df-po 3871  df-so 3872  df-fr 3909  df-we 3911  df-ord 3952  df-on 3953  df-lim 3954  df-suc 3955  df-om 4230
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