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Theorem findes 6701
Description: Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. See tfindes 6668 for the transfinite version. This is an alternative for Metamath 100 proof #74. (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 3327 . 2  |-  ( z  =  (/)  ->  ( [ z  /  x ] ph 
<-> 
[. (/)  /  x ]. ph ) )
2 sbequ 2083 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
3 dfsbcq2 3327 . 2  |-  ( z  =  suc  y  -> 
( [ z  /  x ] ph  <->  [. suc  y  /  x ]. ph )
)
4 sbequ12r 1955 . 2  |-  ( z  =  x  ->  ( [ z  /  x ] ph  <->  ph ) )
5 findes.1 . 2  |-  [. (/)  /  x ]. ph
6 nfv 1678 . . . 4  |-  F/ x  y  e.  om
7 nfs1v 2157 . . . . 5  |-  F/ x [ y  /  x ] ph
8 nfsbc1v 3344 . . . . 5  |-  F/ x [. suc  y  /  x ]. ph
97, 8nfim 1862 . . . 4  |-  F/ x
( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
106, 9nfim 1862 . . 3  |-  F/ x
( y  e.  om  ->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
)
11 eleq1 2532 . . . 4  |-  ( x  =  y  ->  (
x  e.  om  <->  y  e.  om ) )
12 sbequ12 1954 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
13 suceq 4936 . . . . . 6  |-  ( x  =  y  ->  suc  x  =  suc  y )
14 dfsbcq 3326 . . . . . 6  |-  ( suc  x  =  suc  y  ->  ( [. suc  x  /  x ]. ph  <->  [. suc  y  /  x ]. ph )
)
1513, 14syl 16 . . . . 5  |-  ( x  =  y  ->  ( [. suc  x  /  x ]. ph  <->  [. suc  y  /  x ]. ph ) )
1612, 15imbi12d 320 . . . 4  |-  ( x  =  y  ->  (
( ph  ->  [. suc  x  /  x ]. ph )  <->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
) )
1711, 16imbi12d 320 . . 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 1975 . 2  |-  ( y  e.  om  ->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
)
201, 2, 3, 4, 5, 19finds 6697 1  |-  ( x  e.  om  ->  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374   [wsb 1706    e. wcel 1762   [.wsbc 3324   (/)c0 3778   suc csuc 4873   omcom 6671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-tr 4534  df-eprel 4784  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-om 6672
This theorem is referenced by: (None)
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