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Theorem findes 4723
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 4690 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
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3028 . 2  |-  ( z  =  (/)  ->  ( [ z  /  x ] ph 
<-> 
[. (/)  /  x ]. ph ) )
2 sbequ 2032 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
3 dfsbcq2 3028 . 2  |-  ( z  =  suc  y  -> 
( [ z  /  x ] ph  <->  [. suc  y  /  x ]. ph )
)
4 sbequ12r 1892 . 2  |-  ( z  =  x  ->  ( [ z  /  x ] ph  <->  ph ) )
5 findes.1 . 2  |-  [. (/)  /  x ]. ph
6 nfv 1610 . . . 4  |-  F/ x  y  e.  om
7 nfs1v 2078 . . . . 5  |-  F/ x [ y  /  x ] ph
8 nfsbc1v 3044 . . . . 5  |-  F/ x [. suc  y  /  x ]. ph
97, 8nfim 1792 . . . 4  |-  F/ x
( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
106, 9nfim 1792 . . 3  |-  F/ x
( y  e.  om  ->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
)
11 eleq1 2376 . . . 4  |-  ( x  =  y  ->  (
x  e.  om  <->  y  e.  om ) )
12 sbequ12 1891 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
13 suceq 4494 . . . . . 6  |-  ( x  =  y  ->  suc  x  =  suc  y )
14 dfsbcq 3027 . . . . . 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 311 . . . 4  |-  ( x  =  y  ->  (
( ph  ->  [. suc  x  /  x ]. ph )  <->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
) )
1711, 16imbi12d 311 . . 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 1958 . 2  |-  ( y  e.  om  ->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
)
201, 2, 3, 4, 5, 19finds 4719 1  |-  ( x  e.  om  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633   [wsb 1639    e. wcel 1701   [.wsbc 3025   (/)c0 3489   suc csuc 4431   omcom 4693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-tr 4151  df-eprel 4342  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694
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