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Theorem findes 6750
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 6716 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 3282 . 2  |-  ( z  =  (/)  ->  ( [ z  /  x ] ph 
<-> 
[. (/)  /  x ]. ph ) )
2 sbequ 2216 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
3 dfsbcq2 3282 . 2  |-  ( z  =  suc  y  -> 
( [ z  /  x ] ph  <->  [. suc  y  /  x ]. ph )
)
4 sbequ12r 2095 . 2  |-  ( z  =  x  ->  ( [ z  /  x ] ph  <->  ph ) )
5 findes.1 . 2  |-  [. (/)  /  x ]. ph
6 nfv 1772 . . . 4  |-  F/ x  y  e.  om
7 nfs1v 2277 . . . . 5  |-  F/ x [ y  /  x ] ph
8 nfsbc1v 3299 . . . . 5  |-  F/ x [. suc  y  /  x ]. ph
97, 8nfim 2014 . . . 4  |-  F/ x
( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
106, 9nfim 2014 . . 3  |-  F/ x
( y  e.  om  ->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
)
11 eleq1 2528 . . . 4  |-  ( x  =  y  ->  (
x  e.  om  <->  y  e.  om ) )
12 sbequ12 2094 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
13 suceq 5507 . . . . . 6  |-  ( x  =  y  ->  suc  x  =  suc  y )
1413sbceq1d 3284 . . . . 5  |-  ( x  =  y  ->  ( [. suc  x  /  x ]. ph  <->  [. suc  y  /  x ]. ph ) )
1512, 14imbi12d 326 . . . 4  |-  ( x  =  y  ->  (
( ph  ->  [. suc  x  /  x ]. ph )  <->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
) )
1611, 15imbi12d 326 . . 3  |-  ( x  =  y  ->  (
( x  e.  om  ->  ( ph  ->  [. suc  x  /  x ]. ph )
)  <->  ( y  e. 
om  ->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph ) ) ) )
17 findes.2 . . 3  |-  ( x  e.  om  ->  ( ph  ->  [. suc  x  /  x ]. ph ) )
1810, 16, 17chvar 2117 . 2  |-  ( y  e.  om  ->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
)
191, 2, 3, 4, 5, 18finds 6746 1  |-  ( x  e.  om  ->  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   [wsb 1808    e. wcel 1898   [.wsbc 3279   (/)c0 3743   suc csuc 5444   omcom 6719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-tr 4512  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-om 6720
This theorem is referenced by:  rdgeqoa  31818
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