Theorem tfindes 3215

Description: Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction hypothesis for successors, and the third is the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197.

Hypotheses

Ref	Expression
tfindes.1	|- [(/) / x]ph
tfindes.2	|- (x e. On -> (ph -> [suc x / x]ph))
tfindes.3	|- (Lim y -> (A.x e. y ph -> [y / x]ph))

Assertion

Ref	Expression
tfindes	|- (x e. On -> ph)

Distinct variable groups:   x,y   ph,y

Proof of Theorem tfindes

Step	Hyp	Ref	Expression
1		dfsbcq 1988	. 2 |- (y = (/) -> ([y / x]ph <-> [(/) / x]ph))
2		sbequ 1266	. 2 |- (y = z -> ([y / x]ph <-> [z / x]ph))
3		dfsbcq 1988	. 2 |- (y = suc z -> ([y / x]ph <-> [suc z / x]ph))
4		sbequ12r 1219	. 2 |- (y = x -> ([y / x]ph <-> ph))
5		tfindes.1	. 2 |- [(/) / x]ph
6		ax-17 1007	. . . 4 |- (z e. On -> A.x z e. On)
7		hbs1 1371	. . . . 5 |- ([z / x]ph -> A.x[z / x]ph)
8		visset 1859	. . . . . . 7 |- z e. V
9	8	sucex 3168	. . . . . 6 |- suc z e. V
10	9	hbsbc1v 1995	. . . . 5 |- ([suc z / x]ph -> A.x[suc z / x]ph)
11	7, 10	hbim 1043	. . . 4 |- (([z / x]ph -> [suc z / x]ph) -> A.x([z / x]ph -> [suc z / x]ph))
12	6, 11	hbim 1043	. . 3 |- ((z e. On -> ([z / x]ph -> [suc z / x]ph)) -> A.x(z e. On -> ([z / x]ph -> [suc z / x]ph)))
13		eleq1 1577	. . . 4 |- (x = z -> (x e. On <-> z e. On))
14		sbequ12 1218	. . . . 5 |- (x = z -> (ph <-> [z / x]ph))
15		suceq 3038	. . . . . 6 |- (x = z -> suc x = suc z)
16		dfsbcq 1988	. . . . . 6 |- (suc x = suc z -> ([suc x / x]ph <-> [suc z / x]ph))
17	15, 16	syl 10	. . . . 5 |- (x = z -> ([suc x / x]ph <-> [suc z / x]ph))
18	14, 17	imbi12d 629	. . . 4 |- (x = z -> ((ph -> [suc x / x]ph) <-> ([z / x]ph -> [suc z / x]ph)))
19	13, 18	imbi12d 629	. . 3 |- (x = z -> ((x e. On -> (ph -> [suc x / x]ph)) <-> (z e. On -> ([z / x]ph -> [suc z / x]ph))))
20		tfindes.2	. . 3 |- (x e. On -> (ph -> [suc x / x]ph))
21	12, 19, 20	chvar 1204	. 2 |- (z e. On -> ([z / x]ph -> [suc z / x]ph))
22		tfindes.3	. . 3 |- (Lim y -> (A.x e. y ph -> [y / x]ph))
23		ax-17 1007	. . . 4 |- (ph -> A.zph)
24	23, 7, 14	cbvral 1844	. . 3 |- (A.x e. y ph <-> A.z e. y [z / x]ph)
25	22, 24	syl5ibr 205	. 2 |- (Lim y -> (A.z e. y [z / x]ph -> [y / x]ph))
26	1, 2, 3, 4, 5, 21, 25	tfinds 3212	1 |- (x e. On -> ph)

Syntax hints:   -> wi 3   <-> wb 144   = wceq 992   e. wcel 994  [wsbc 1207  A.wral 1691  (/)c0 2332  Oncon0 2975  Lim wlim 2976  suc csuc 2977

This theorem is referenced by:  tfinds2 3216

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-rab 1698  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

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