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Theorem findes 3984
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 3950 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 2505 . 2 |- (z = (/) -> ([z / x]ph <-> [(/) / x]ph))
2 sbequ 1664 . 2 |- (z = y -> ([z / x]ph <-> [y / x]ph))
3 dfsbcq 2505 . 2 |- (z = suc y -> ([z / x]ph <-> [suc y / x]ph))
4 sbequ12r 1615 . 2 |- (z = x -> ([z / x]ph <-> ph))
5 findes.1 . 2 |- [(/) / x]ph
6 ax-17 1444 . . . 4 |- (y e. om -> A.x y e. om)
7 hbs1 1774 . . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
8 vex 2340 . . . . . . 7 |- y e. _V
98sucex 3903 . . . . . 6 |- suc y e. _V
109hbsbc1v 2512 . . . . 5 |- ([suc y / x]ph -> A.x[suc y / x]ph)
117, 10hbim 1481 . . . 4 |- (([y / x]ph -> [suc y / x]ph) -> A.x([y / x]ph -> [suc y / x]ph))
126, 11hbim 1481 . . 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 1991 . . . 4 |- (x = y -> (x e. om <-> y e. om))
14 sbequ12 1614 . . . . 5 |- (x = y -> (ph <-> [y / x]ph))
15 suceq 3738 . . . . . 6 |- (x = y -> suc x = suc y)
16 dfsbcq 2505 . . . . . 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 308 . . . 4 |- (x = y -> ((ph -> [suc x / x]ph) <-> ([y / x]ph -> [suc y / x]ph)))
1913, 18imbi12d 308 . . 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 1600 . 2 |- (y e. om -> ([y / x]ph -> [suc y / x]ph))
221, 2, 3, 4, 5, 21finds 3980 1 |- (x e. om -> ph)
Colors of variables: wff set class
Syntax hints:   -> wi 4   <-> wb 174   = wceq 1428   e. wcel 1430  [wsbc 1603  (/)c0 2900  suc csuc 3676  omcom 3953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1345  ax-6 1346  ax-7 1347  ax-gen 1348  ax-8 1432  ax-10 1433  ax-11 1434  ax-12 1435  ax-13 1436  ax-14 1437  ax-17 1444  ax-9 1459  ax-4 1465  ax-16 1643  ax-ext 1914  ax-sep 3450  ax-nul 3459  ax-pr 3519  ax-un 3791
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-3or 916  df-3an 917  df-ex 1350  df-sb 1605  df-eu 1832  df-mo 1833  df-clab 1920  df-cleq 1925  df-clel 1928  df-ne 2052  df-ral 2145  df-rex 2146  df-rab 2148  df-v 2339  df-sbc 2504  df-dif 2639  df-un 2641  df-in 2643  df-ss 2645  df-pss 2647  df-nul 2901  df-if 3002  df-pw 3060  df-sn 3077  df-pr 3078  df-tp 3079  df-op 3080  df-uni 3210  df-br 3355  df-opab 3409  df-tr 3424  df-eprel 3604  df-po 3612  df-so 3626  df-fr 3645  df-we 3661  df-ord 3677  df-on 3678  df-lim 3679  df-suc 3680  df-om 3954
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