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Theorem findes 6609
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 6576 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 3290 . 2  |-  ( z  =  (/)  ->  ( [ z  /  x ] ph 
<-> 
[. (/)  /  x ]. ph ) )
2 sbequ 2074 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
3 dfsbcq2 3290 . 2  |-  ( z  =  suc  y  -> 
( [ z  /  x ] ph  <->  [. suc  y  /  x ]. ph )
)
4 sbequ12r 1946 . 2  |-  ( z  =  x  ->  ( [ z  /  x ] ph  <->  ph ) )
5 findes.1 . 2  |-  [. (/)  /  x ]. ph
6 nfv 1674 . . . 4  |-  F/ x  y  e.  om
7 nfs1v 2149 . . . . 5  |-  F/ x [ y  /  x ] ph
8 nfsbc1v 3307 . . . . 5  |-  F/ x [. suc  y  /  x ]. ph
97, 8nfim 1855 . . . 4  |-  F/ x
( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
106, 9nfim 1855 . . 3  |-  F/ x
( y  e.  om  ->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
)
11 eleq1 2523 . . . 4  |-  ( x  =  y  ->  (
x  e.  om  <->  y  e.  om ) )
12 sbequ12 1945 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
13 suceq 4885 . . . . . 6  |-  ( x  =  y  ->  suc  x  =  suc  y )
14 dfsbcq 3289 . . . . . 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 1966 . 2  |-  ( y  e.  om  ->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
)
201, 2, 3, 4, 5, 19finds 6605 1  |-  ( x  e.  om  ->  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370   [wsb 1702    e. wcel 1758   [.wsbc 3287   (/)c0 3738   suc csuc 4822   omcom 6579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-tr 4487  df-eprel 4733  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-om 6580
This theorem is referenced by: (None)
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