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Theorem findes 4005
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 3971 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 2534 . 2 |- (z = (/) -> ([z / x]ph <-> [(/) / x]ph))
2 sbequ 1693 . 2 |- (z = y -> ([z / x]ph <-> [y / x]ph))
3 dfsbcq 2534 . 2 |- (z = suc y -> ([z / x]ph <-> [suc y / x]ph))
4 sbequ12r 1644 . 2 |- (z = x -> ([z / x]ph <-> ph))
5 findes.1 . 2 |- [(/) / x]ph
6 ax-17 1473 . . . 4 |- (y e. om -> A.x y e. om)
7 hbs1 1803 . . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
8 vex 2369 . . . . . . 7 |- y e. _V
98sucex 3924 . . . . . 6 |- suc y e. _V
109hbsbc1v 2541 . . . . 5 |- ([suc y / x]ph -> A.x[suc y / x]ph)
117, 10hbim 1510 . . . 4 |- (([y / x]ph -> [suc y / x]ph) -> A.x([y / x]ph -> [suc y / x]ph))
126, 11hbim 1510 . . 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 2020 . . . 4 |- (x = y -> (x e. om <-> y e. om))
14 sbequ12 1643 . . . . 5 |- (x = y -> (ph <-> [y / x]ph))
15 suceq 3761 . . . . . 6 |- (x = y -> suc x = suc y)
16 dfsbcq 2534 . . . . . 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 330 . . . 4 |- (x = y -> ((ph -> [suc x / x]ph) <-> ([y / x]ph -> [suc y / x]ph)))
1913, 18imbi12d 330 . . 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 1629 . 2 |- (y e. om -> ([y / x]ph -> [suc y / x]ph))
221, 2, 3, 4, 5, 21finds 4001 1 |- (x e. om -> ph)
Colors of variables: wff set class
Syntax hints:   -> wi 4   <-> wb 189   = wceq 1457   e. wcel 1459  [wsbc 1632  (/)c0 2927  suc csuc 3699  omcom 3974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1376  ax-6 1377  ax-7 1378  ax-gen 1379  ax-8 1461  ax-10 1462  ax-11 1463  ax-12 1464  ax-13 1465  ax-14 1466  ax-17 1473  ax-9 1488  ax-4 1494  ax-16 1672  ax-ext 1943  ax-sep 3475  ax-nul 3484  ax-pr 3544  ax-un 3814
This theorem depends on definitions:  df-bi 190  df-or 383  df-an 384  df-3or 947  df-3an 948  df-ex 1381  df-sb 1634  df-eu 1861  df-mo 1862  df-clab 1949  df-cleq 1954  df-clel 1957  df-ne 2081  df-ral 2174  df-rex 2175  df-rab 2177  df-v 2368  df-sbc 2533  df-dif 2666  df-un 2668  df-in 2670  df-ss 2672  df-pss 2674  df-nul 2928  df-if 3029  df-pw 3087  df-sn 3102  df-pr 3103  df-tp 3105  df-op 3106  df-uni 3235  df-br 3380  df-opab 3434  df-tr 3449  df-eprel 3627  df-po 3635  df-so 3649  df-fr 3668  df-we 3684  df-ord 3700  df-on 3701  df-lim 3702  df-suc 3703  df-om 3975
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