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Theorem findes 4272
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 4239 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 2669 . 2  |-  ( z  =  (/)  ->  ( [
z  /  x ] ph 
<->  [ (/)  /  x ] ph ) )
2 sbequ 1816 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph 
<->  [ y  /  x ] ph ) )
3 dfsbcq 2669 . 2  |-  ( z  =  suc  y  ->  ( [ z  /  x ] ph  <->  [ suc  y  /  x ] ph ) )
4 sbequ12r 1764 . 2  |-  ( z  =  x  ->  ( [ z  /  x ] ph 
<-> 
ph ) )
5 findes.1 . 2  |-  [ (/)  /  x ] ph
6 ax-17 1545 . . . 4  |-  ( y  e.  om  ->  A. x  y  e.  om )
7 hbs1 1927 . . . . 5  |-  ( [
y  /  x ] ph  ->  A. x [ y  /  x ] ph )
8 hbsbc1 2677 . . . . 5  |-  ( [ suc  y  /  x ] ph  ->  A. x [ suc  y  /  x ] ph )
97, 8hbim 1622 . . . 4  |-  ( ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph )  ->  A. x ( [ y  /  x ] ph  ->  [ suc  y  /  x ] ph )
)
106, 9hbim 1622 . . 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 2141 . . . 4  |-  ( x  =  y  ->  ( x  e.  om  <->  y  e.  om ) )
12 sbequ12 1763 . . . . 5  |-  ( x  =  y  ->  ( ph  <->  [ y  /  x ] ph ) )
13 suceq 4022 . . . . . 6  |-  ( x  =  y  ->  suc  x  =  suc  y )
14 dfsbcq 2669 . . . . . 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 1748 . 2  |-  ( y  e.  om  ->  ( [
y  /  x ] ph  ->  [ suc  y  /  x ] ph )
)
201, 2, 3, 4, 5, 19finds 4268 1  |-  ( x  e.  om  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 174    = wceq 1536    e. wcel 1538   [wsbc 1752   (/)c0 3088   suc csuc 3959   omcom 4242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1451  ax-6 1452  ax-7 1453  ax-gen 1454  ax-8 1540  ax-11 1541  ax-13 1542  ax-14 1543  ax-17 1545  ax-12o 1578  ax-10 1592  ax-9 1598  ax-4 1606  ax-16 1793  ax-ext 2064  ax-sep 3715  ax-nul 3723  ax-pr 3783  ax-un 4075
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-3or 904  df-3an 905  df-ex 1456  df-sb 1754  df-eu 1976  df-mo 1977  df-clab 2070  df-cleq 2075  df-clel 2078  df-ne 2201  df-ral 2295  df-rex 2296  df-rab 2298  df-v 2494  df-sbc 2668  df-dif 2813  df-un 2815  df-in 2817  df-ss 2821  df-pss 2823  df-nul 3089  df-if 3199  df-pw 3260  df-sn 3278  df-pr 3279  df-tp 3280  df-op 3281  df-uni 3439  df-br 3601  df-opab 3655  df-tr 3688  df-eprel 3870  df-po 3879  df-so 3880  df-fr 3917  df-we 3919  df-ord 3960  df-on 3961  df-lim 3962  df-suc 3963  df-om 4243
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