MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  findes Unicode version

Theorem findes 4685
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 4652 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 2995 . 2  |-  ( z  =  (/)  ->  ( [ z  /  x ] ph 
<-> 
[. (/)  /  x ]. ph ) )
2 sbequ 1999 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
3 dfsbcq2 2995 . 2  |-  ( z  =  suc  y  -> 
( [ z  /  x ] ph  <->  [. suc  y  /  x ]. ph )
)
4 sbequ12r 1863 . 2  |-  ( z  =  x  ->  ( [ z  /  x ] ph  <->  ph ) )
5 findes.1 . 2  |-  [. (/)  /  x ]. ph
6 nfv 1605 . . . 4  |-  F/ x  y  e.  om
7 nfs1v 2045 . . . . 5  |-  F/ x [ y  /  x ] ph
8 nfsbc1v 3011 . . . . 5  |-  F/ x [. suc  y  /  x ]. ph
97, 8nfim 1771 . . . 4  |-  F/ x
( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
106, 9nfim 1771 . . 3  |-  F/ x
( y  e.  om  ->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
)
11 eleq1 2344 . . . 4  |-  ( x  =  y  ->  (
x  e.  om  <->  y  e.  om ) )
12 sbequ12 1862 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
13 suceq 4456 . . . . . 6  |-  ( x  =  y  ->  suc  x  =  suc  y )
14 dfsbcq 2994 . . . . . 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 1929 . 2  |-  ( y  e.  om  ->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
)
201, 2, 3, 4, 5, 19finds 4681 1  |-  ( x  e.  om  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   [wsb 1630    e. wcel 1685   [.wsbc 2992   (/)c0 3456   suc csuc 4393   omcom 4655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656
  Copyright terms: Public domain W3C validator