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Theorem findes 3992
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 3958 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 2511 . 2 |- (z = (/) -> ([z / x]ph <-> [(/) / x]ph))
2 sbequ 1670 . 2 |- (z = y -> ([z / x]ph <-> [y / x]ph))
3 dfsbcq 2511 . 2 |- (z = suc y -> ([z / x]ph <-> [suc y / x]ph))
4 sbequ12r 1621 . 2 |- (z = x -> ([z / x]ph <-> ph))
5 findes.1 . 2 |- [(/) / x]ph
6 ax-17 1450 . . . 4 |- (y e. om -> A.x y e. om)
7 hbs1 1780 . . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
8 vex 2346 . . . . . . 7 |- y e. _V
98sucex 3911 . . . . . 6 |- suc y e. _V
109hbsbc1v 2518 . . . . 5 |- ([suc y / x]ph -> A.x[suc y / x]ph)
117, 10hbim 1487 . . . 4 |- (([y / x]ph -> [suc y / x]ph) -> A.x([y / x]ph -> [suc y / x]ph))
126, 11hbim 1487 . . 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 1997 . . . 4 |- (x = y -> (x e. om <-> y e. om))
14 sbequ12 1620 . . . . 5 |- (x = y -> (ph <-> [y / x]ph))
15 suceq 3746 . . . . . 6 |- (x = y -> suc x = suc y)
16 dfsbcq 2511 . . . . . 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 311 . . . 4 |- (x = y -> ((ph -> [suc x / x]ph) <-> ([y / x]ph -> [suc y / x]ph)))
1913, 18imbi12d 311 . . 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 1606 . 2 |- (y e. om -> ([y / x]ph -> [suc y / x]ph))
221, 2, 3, 4, 5, 21finds 3988 1 |- (x e. om -> ph)
Colors of variables: wff set class
Syntax hints:   -> wi 4   <-> wb 174   = wceq 1434   e. wcel 1436  [wsbc 1609  (/)c0 2906  suc csuc 3684  omcom 3961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1351  ax-6 1352  ax-7 1353  ax-gen 1354  ax-8 1438  ax-10 1439  ax-11 1440  ax-12 1441  ax-13 1442  ax-14 1443  ax-17 1450  ax-9 1465  ax-4 1471  ax-16 1649  ax-ext 1920  ax-sep 3458  ax-nul 3467  ax-pr 3527  ax-un 3799
This theorem depends on definitions:  df-bi 175  df-or 362  df-an 363  df-3or 922  df-3an 923  df-ex 1356  df-sb 1611  df-eu 1838  df-mo 1839  df-clab 1926  df-cleq 1931  df-clel 1934  df-ne 2058  df-ral 2151  df-rex 2152  df-rab 2154  df-v 2345  df-sbc 2510  df-dif 2645  df-un 2647  df-in 2649  df-ss 2651  df-pss 2653  df-nul 2907  df-if 3009  df-pw 3067  df-sn 3084  df-pr 3085  df-tp 3086  df-op 3087  df-uni 3218  df-br 3363  df-opab 3417  df-tr 3432  df-eprel 3612  df-po 3620  df-so 3634  df-fr 3653  df-we 3669  df-ord 3685  df-on 3686  df-lim 3687  df-suc 3688  df-om 3962
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