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Theorem findes 4001
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 3967 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
StepHypRef Expression
1 dfsbcq 2526 . 2 |- (z = (/) -> ([z / x]ph <-> [(/) / x]ph))
2 sbequ 1685 . 2 |- (z = y -> ([z / x]ph <-> [y / x]ph))
3 dfsbcq 2526 . 2 |- (z = suc y -> ([z / x]ph <-> [suc y / x]ph))
4 sbequ12r 1636 . 2 |- (z = x -> ([z / x]ph <-> ph))
5 findes.1 . 2 |- [(/) / x]ph
6 ax-17 1465 . . . 4 |- (y e. om -> A.x y e. om)
7 hbs1 1795 . . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
8 vex 2361 . . . . . . 7 |- y e. _V
98sucex 3920 . . . . . 6 |- suc y e. _V
109hbsbc1v 2533 . . . . 5 |- ([suc y / x]ph -> A.x[suc y / x]ph)
117, 10hbim 1502 . . . 4 |- (([y / x]ph -> [suc y / x]ph) -> A.x([y / x]ph -> [suc y / x]ph))
126, 11hbim 1502 . . 3 |- ((y e. om -> ([y / x]ph -> [suc y / x]ph)) -> A.x(y e. om -> ([y / x]ph -> [suc y / x]ph)))
13 eleq1 2012 . . . 4 |- (x = y -> (x e. om <-> y e. om))
14 sbequ12 1635 . . . . 5 |- (x = y -> (ph <-> [y / x]ph))
15 suceq 3755 . . . . . 6 |- (x = y -> suc x = suc y)
16 dfsbcq 2526 . . . . . 6 |- (suc x = suc y -> ([suc x / x]ph <-> [suc y / x]ph))
1715, 16syl 14 . . . . 5 |- (x = y -> ([suc x / x]ph <-> [suc y / x]ph))
1814, 17imbi12d 325 . . . 4 |- (x = y -> ((ph -> [suc x / x]ph) <-> ([y / x]ph -> [suc y / x]ph)))
1913, 18imbi12d 325 . . 3 |- (x = y -> ((x e. om -> (ph -> [suc x / x]ph)) <-> (y e. om -> ([y / x]ph -> [suc y / x]ph))))
20 findes.2 . . 3 |- (x e. om -> (ph -> [suc x / x]ph))
2112, 19, 20chvar 1621 . 2 |- (y e. om -> ([y / x]ph -> [suc y / x]ph))
221, 2, 3, 4, 5, 21finds 3997 1 |- (x e. om -> ph)
Colors of variables: wff set class
Syntax hints:   -> wi 4   <-> wb 184   = wceq 1449   e. wcel 1451  [wsbc 1624  (/)c0 2921  suc csuc 3693  omcom 3970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1367  ax-6 1368  ax-7 1369  ax-gen 1370  ax-8 1453  ax-10 1454  ax-11 1455  ax-12 1456  ax-13 1457  ax-14 1458  ax-17 1465  ax-9 1480  ax-4 1486  ax-16 1664  ax-ext 1935  ax-sep 3469  ax-nul 3478  ax-pr 3538  ax-un 3808
This theorem depends on definitions:  df-bi 185  df-or 378  df-an 379  df-3or 938  df-3an 939  df-ex 1372  df-sb 1626  df-eu 1853  df-mo 1854  df-clab 1941  df-cleq 1946  df-clel 1949  df-ne 2073  df-ral 2166  df-rex 2167  df-rab 2169  df-v 2360  df-sbc 2525  df-dif 2660  df-un 2662  df-in 2664  df-ss 2666  df-pss 2668  df-nul 2922  df-if 3023  df-pw 3081  df-sn 3096  df-pr 3097  df-tp 3099  df-op 3100  df-uni 3229  df-br 3374  df-opab 3428  df-tr 3443  df-eprel 3621  df-po 3629  df-so 3643  df-fr 3662  df-we 3678  df-ord 3694  df-on 3695  df-lim 3696  df-suc 3697  df-om 3971
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