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Theorem findes 4134
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 4100 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 2672 . 2 |- (z = (/) -> ([z / x]ph <-> [(/) / x]ph))
2 sbequ 1824 . 2 |- (z = y -> ([z / x]ph <-> [y / x]ph))
3 dfsbcq 2672 . 2 |- (z = suc y -> ([z / x]ph <-> [suc y / x]ph))
4 sbequ12r 1775 . 2 |- (z = x -> ([z / x]ph <-> ph))
5 findes.1 . 2 |- [(/) / x]ph
6 ax-17 1608 . . . 4 |- (y e. om -> A.x y e. om)
7 hbs1 1934 . . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
8 visset 2502 . . . . . . 7 |- y e. _V
98sucex 4053 . . . . . 6 |- suc y e. _V
109hbsbc1v 2679 . . . . 5 |- ([suc y / x]ph -> A.x[suc y / x]ph)
117, 10hbim 1642 . . . 4 |- (([y / x]ph -> [suc y / x]ph) -> A.x([y / x]ph -> [suc y / x]ph))
126, 11hbim 1642 . . 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 2155 . . . 4 |- (x = y -> (x e. om <-> y e. om))
14 sbequ12 1774 . . . . 5 |- (x = y -> (ph <-> [y / x]ph))
15 suceq 3893 . . . . . 6 |- (x = y -> suc x = suc y)
16 dfsbcq 2672 . . . . . 6 |- (suc x = suc y -> ([suc x / x]ph <-> [suc y / x]ph))
1715, 16syl 13 . . . . 5 |- (x = y -> ([suc x / x]ph <-> [suc y / x]ph))
1814, 17imbi12d 364 . . . 4 |- (x = y -> ((ph -> [suc x / x]ph) <-> ([y / x]ph -> [suc y / x]ph)))
1913, 18imbi12d 364 . . 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 1760 . 2 |- (y e. om -> ([y / x]ph -> [suc y / x]ph))
221, 2, 3, 4, 5, 21finds 4130 1 |- (x e. om -> ph)
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 209   = wceq 1592   e. wcel 1594  [wsbc 1763  (/)c0 3065  suc csuc 3831  omcom 4103
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1516  ax-6 1517  ax-7 1518  ax-gen 1519  ax-8 1596  ax-10 1597  ax-11 1598  ax-12 1599  ax-13 1600  ax-14 1601  ax-17 1608  ax-9 1620  ax-4 1626  ax-16 1803  ax-ext 2074  ax-sep 3609  ax-nul 3619  ax-pr 3679  ax-un 3947
This theorem depends on definitions:  df-bi 210  df-or 419  df-an 420  df-3or 1038  df-3an 1039  df-ex 1521  df-sb 1765  df-eu 1992  df-mo 1993  df-clab 2080  df-cleq 2085  df-clel 2088  df-ne 2220  df-ral 2314  df-rex 2315  df-rab 2317  df-v 2501  df-sbc 2671  df-dif 2804  df-un 2806  df-in 2808  df-ss 2810  df-pss 2812  df-nul 3066  df-if 3166  df-pw 3222  df-sn 3237  df-pr 3238  df-tp 3240  df-op 3241  df-uni 3365  df-br 3510  df-opab 3568  df-tr 3583  df-eprel 3762  df-po 3770  df-so 3782  df-fr 3800  df-we 3816  df-ord 3832  df-on 3833  df-lim 3834  df-suc 3835  df-om 4104
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