Theorem findes 2400

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. (Contributed by Raph Levien, 9-Jul-03.)

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 1442	. 2 |- (z = (/) -> ([z / x]ph <-> [(/) / x]ph))
2		sbequ 877	. 2 |- (z = y -> ([z / x]ph <-> [y / x]ph))
3		dfsbcq 1442	. 2 |- (z = suc y -> ([z / x]ph <-> [suc y / x]ph))
4		sbequ12r 866	. 2 |- (z = x -> ([z / x]ph <-> ph))
5		findes.1	. 2 |- [(/) / x]ph
6		ax-17 925	. . . 4 |- (y e. om -> A.x y e. om)
7		hbs1 986	. . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
8		visset 1350	. . . . . . 7 |- y e. V
9	8	sucex 2303	. . . . . 6 |- suc y e. V
10	9	hbsbcv 1447	. . . . 5 |- ([suc y / x]ph -> A.x[suc y / x]ph)
11	7, 10	hbim 702	. . . 4 |- (([y / x]ph -> [suc y / x]ph) -> A.x([y / x]ph -> [suc y / x]ph))
12	6, 11	hbim 702	. . 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 1149	. . . 4 |- (x = y -> (x e. om <-> y e. om))
14		sbequ12 865	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
15		suceq 2288	. . . . . 6 |- (x = y -> suc x = suc y)
16		dfsbcq 1442	. . . . . 6 |- (suc x = suc y -> ([suc x / x]ph <-> [suc y / x]ph))
17	15, 16	syl 12	. . . . 5 |- (x = y -> ([suc x / x]ph <-> [suc y / x]ph))
18	14, 17	imbi12d 474	. . . 4 |- (x = y -> ((ph -> [suc x / x]ph) <-> ([y / x]ph -> [suc y / x]ph)))
19	13, 18	imbi12d 474	. . 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	chv2 850	. 2 |- (y e. om -> ([y / x]ph -> [suc y / x]ph))
22	1, 2, 3, 4, 5, 21	finds 2397	1 |- (x e. om -> ph)

Syntax hints:   -> wi 2   <-> wb 127   = weq 797  [wsb 852   = wceq 1091   e. wcel 1092  [wsbc 1440  (/)c0 1707  suc csuc 2201  omcom 2372

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373

metamath.org