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Theorem findes 4117
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 4083 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 2655 . 2 |- (z = (/) -> ([z / x]ph <-> [(/) / x]ph))
2 sbequ 1807 . 2 |- (z = y -> ([z / x]ph <-> [y / x]ph))
3 dfsbcq 2655 . 2 |- (z = suc y -> ([z / x]ph <-> [suc y / x]ph))
4 sbequ12r 1758 . 2 |- (z = x -> ([z / x]ph <-> ph))
5 findes.1 . 2 |- [(/) / x]ph
6 ax-17 1589 . . . 4 |- (y e. om -> A.x y e. om)
7 hbs1 1917 . . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
8 visset 2485 . . . . . . 7 |- y e. _V
98sucex 4036 . . . . . 6 |- suc y e. _V
109hbsbc1v 2662 . . . . 5 |- ([suc y / x]ph -> A.x[suc y / x]ph)
117, 10hbim 1625 . . . 4 |- (([y / x]ph -> [suc y / x]ph) -> A.x([y / x]ph -> [suc y / x]ph))
126, 11hbim 1625 . . 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 2138 . . . 4 |- (x = y -> (x e. om <-> y e. om))
14 sbequ12 1757 . . . . 5 |- (x = y -> (ph <-> [y / x]ph))
15 suceq 3876 . . . . . 6 |- (x = y -> suc x = suc y)
16 dfsbcq 2655 . . . . . 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 358 . . . 4 |- (x = y -> ((ph -> [suc x / x]ph) <-> ([y / x]ph -> [suc y / x]ph)))
1913, 18imbi12d 358 . . 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 1743 . 2 |- (y e. om -> ([y / x]ph -> [suc y / x]ph))
221, 2, 3, 4, 5, 21finds 4113 1 |- (x e. om -> ph)
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 203   = wceq 1573   e. wcel 1575  [wsbc 1746  (/)c0 3048  suc csuc 3814  omcom 4086
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1497  ax-6 1498  ax-7 1499  ax-gen 1500  ax-8 1577  ax-10 1578  ax-11 1579  ax-12 1580  ax-13 1581  ax-14 1582  ax-17 1589  ax-9 1603  ax-4 1609  ax-16 1786  ax-ext 2057  ax-sep 3592  ax-nul 3602  ax-pr 3662  ax-un 3930
This theorem depends on definitions:  df-bi 204  df-or 413  df-an 414  df-3or 1019  df-3an 1020  df-ex 1502  df-sb 1748  df-eu 1975  df-mo 1976  df-clab 2063  df-cleq 2068  df-clel 2071  df-ne 2203  df-ral 2297  df-rex 2298  df-rab 2300  df-v 2484  df-sbc 2654  df-dif 2787  df-un 2789  df-in 2791  df-ss 2793  df-pss 2795  df-nul 3049  df-if 3149  df-pw 3205  df-sn 3220  df-pr 3221  df-tp 3223  df-op 3224  df-uni 3348  df-br 3493  df-opab 3551  df-tr 3566  df-eprel 3745  df-po 3753  df-so 3765  df-fr 3783  df-we 3799  df-ord 3815  df-on 3816  df-lim 3817  df-suc 3818  df-om 4087
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