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Theorem findes 4251
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 4218 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 2650 . 2  |-  ( z  =  (/)  ->  ( [ z  /  x ] ph 
<->  [ (/)  /  x ] ph ) )
2 sbequ 1796 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
3 dfsbcq 2650 . 2  |-  ( z  =  suc  y  -> 
( [ z  /  x ] ph  <->  [ suc  y  /  x ] ph ) )
4 sbequ12r 1744 . 2  |-  ( z  =  x  ->  ( [ z  /  x ] ph  <->  ph ) )
5 findes.1 . 2  |-  [ (/)  /  x ] ph
6 ax-17 1527 . . . 4  |-  ( y  e.  om  ->  A. x  y  e.  om )
7 hbs1 1907 . . . . 5  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
8 hbsbc1 2660 . . . . 5  |-  ( [ suc  y  /  x ] ph  ->  A. x [ suc  y  /  x ] ph )
97, 8hbim 1603 . . . 4  |-  ( ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph )  ->  A. x ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph )
)
106, 9hbim 1603 . . 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 2122 . . . 4  |-  ( x  =  y  ->  (
x  e.  om  <->  y  e.  om ) )
12 sbequ12 1743 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
13 suceq 4006 . . . . . 6  |-  ( x  =  y  ->  suc  x  =  suc  y )
14 dfsbcq 2650 . . . . . 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 1728 . 2  |-  ( y  e.  om  ->  ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph ) )
201, 2, 3, 4, 5, 19finds 4247 1  |-  ( x  e.  om  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 174    = wceq 1518    e. wcel 1520   [wsbc 1732   (/)c0 3070   suc csuc 3943   omcom 4221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1440  ax-6 1441  ax-7 1442  ax-gen 1443  ax-8 1522  ax-11 1523  ax-13 1524  ax-14 1525  ax-17 1527  ax-12o 1560  ax-10 1574  ax-9 1580  ax-4 1587  ax-16 1773  ax-ext 2044  ax-sep 3699  ax-nul 3707  ax-pr 3767  ax-un 4059
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 895  df-3an 896  df-ex 1445  df-sb 1734  df-eu 1956  df-mo 1957  df-clab 2050  df-cleq 2055  df-clel 2058  df-ne 2182  df-ral 2276  df-rex 2277  df-rab 2279  df-v 2475  df-sbc 2649  df-dif 2794  df-un 2796  df-in 2798  df-ss 2802  df-pss 2804  df-nul 3071  df-if 3180  df-pw 3241  df-sn 3259  df-pr 3260  df-tp 3261  df-op 3262  df-uni 3423  df-br 3585  df-opab 3639  df-tr 3672  df-eprel 3854  df-po 3863  df-so 3864  df-fr 3901  df-we 3903  df-ord 3944  df-on 3945  df-lim 3946  df-suc 3947  df-om 4222
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