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Theorem findes 4250
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 4217 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 2649 . 2  |-  ( z  =  (/)  ->  ( [ z  /  x ] ph 
<->  [ (/)  /  x ] ph ) )
2 sbequ 1795 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
3 dfsbcq 2649 . 2  |-  ( z  =  suc  y  -> 
( [ z  /  x ] ph  <->  [ suc  y  /  x ] ph ) )
4 sbequ12r 1743 . 2  |-  ( z  =  x  ->  ( [ z  /  x ] ph  <->  ph ) )
5 findes.1 . 2  |-  [ (/)  /  x ] ph
6 ax-17 1526 . . . 4  |-  ( y  e.  om  ->  A. x  y  e.  om )
7 hbs1 1906 . . . . 5  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
8 hbsbc1 2657 . . . . 5  |-  ( [ suc  y  /  x ] ph  ->  A. x [ suc  y  /  x ] ph )
97, 8hbim 1602 . . . 4  |-  ( ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph )  ->  A. x ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph )
)
106, 9hbim 1602 . . 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 2121 . . . 4  |-  ( x  =  y  ->  (
x  e.  om  <->  y  e.  om ) )
12 sbequ12 1742 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
13 suceq 4005 . . . . . 6  |-  ( x  =  y  ->  suc  x  =  suc  y )
14 dfsbcq 2649 . . . . . 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 1727 . 2  |-  ( y  e.  om  ->  ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph ) )
201, 2, 3, 4, 5, 19finds 4246 1  |-  ( x  e.  om  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 174    = wceq 1517    e. wcel 1519   [wsbc 1731   (/)c0 3069   suc csuc 3942   omcom 4220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1439  ax-6 1440  ax-7 1441  ax-gen 1442  ax-8 1521  ax-11 1522  ax-13 1523  ax-14 1524  ax-17 1526  ax-12o 1559  ax-10 1573  ax-9 1579  ax-4 1586  ax-16 1772  ax-ext 2043  ax-sep 3698  ax-nul 3706  ax-pr 3766  ax-un 4058
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 894  df-3an 895  df-ex 1444  df-sb 1733  df-eu 1955  df-mo 1956  df-clab 2049  df-cleq 2054  df-clel 2057  df-ne 2181  df-ral 2275  df-rex 2276  df-rab 2278  df-v 2474  df-sbc 2648  df-dif 2793  df-un 2795  df-in 2797  df-ss 2801  df-pss 2803  df-nul 3070  df-if 3179  df-pw 3240  df-sn 3258  df-pr 3259  df-tp 3260  df-op 3261  df-uni 3422  df-br 3584  df-opab 3638  df-tr 3671  df-eprel 3853  df-po 3862  df-so 3863  df-fr 3900  df-we 3902  df-ord 3943  df-on 3944  df-lim 3945  df-suc 3946  df-om 4221
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