Theorem findes 3248

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 3215 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

Step	Hyp	Ref	Expression
1		dfsbcq 1988	. 2 |- (z = (/) -> ([z / x]ph <-> [(/) / x]ph))
2		sbequ 1266	. 2 |- (z = y -> ([z / x]ph <-> [y / x]ph))
3		dfsbcq 1988	. 2 |- (z = suc y -> ([z / x]ph <-> [suc y / x]ph))
4		sbequ12r 1219	. 2 |- (z = x -> ([z / x]ph <-> ph))
5		findes.1	. 2 |- [(/) / x]ph
6		ax-17 1007	. . . 4 |- (y e. om -> A.x y e. om)
7		hbs1 1371	. . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
8		visset 1859	. . . . . . 7 |- y e. V
9	8	sucex 3168	. . . . . 6 |- suc y e. V
10	9	hbsbc1v 1995	. . . . 5 |- ([suc y / x]ph -> A.x[suc y / x]ph)
11	7, 10	hbim 1043	. . . 4 |- (([y / x]ph -> [suc y / x]ph) -> A.x([y / x]ph -> [suc y / x]ph))
12	6, 11	hbim 1043	. . 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 1577	. . . 4 |- (x = y -> (x e. om <-> y e. om))
14		sbequ12 1218	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
15		suceq 3038	. . . . . 6 |- (x = y -> suc x = suc y)
16		dfsbcq 1988	. . . . . 6 |- (suc x = suc y -> ([suc x / x]ph <-> [suc y / x]ph))
17	15, 16	syl 10	. . . . 5 |- (x = y -> ([suc x / x]ph <-> [suc y / x]ph))
18	14, 17	imbi12d 629	. . . 4 |- (x = y -> ((ph -> [suc x / x]ph) <-> ([y / x]ph -> [suc y / x]ph)))
19	13, 18	imbi12d 629	. . 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))
21	12, 19, 20	chvar 1204	. 2 |- (y e. om -> ([y / x]ph -> [suc y / x]ph))
22	1, 2, 3, 4, 5, 21	finds 3244	1 |- (x e. om -> ph)

Syntax hints:   -> wi 3   <-> wb 144   = wceq 992   e. wcel 994  [wsbc 1207  (/)c0 2332  suc csuc 2977  omcom 3218

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-sbc 1987  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219

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