Theorem tfindes 3946

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 2455	. 2 |- (y = (/) -> ([y / x]ph <-> [(/) / x]ph))
2		sbequ 1599	. 2 |- (y = z -> ([y / x]ph <-> [z / x]ph))
3		dfsbcq 2455	. 2 |- (y = suc z -> ([y / x]ph <-> [suc z / x]ph))
4		sbequ12r 1546	. 2 |- (y = x -> ([y / x]ph <-> ph))
5		tfindes.1	. 2 |- [(/) / x]ph
6		ax-17 1317	. . . 4 |- (z e. On -> A.x z e. On)
7		hbs1 1722	. . . . 5 |- ([z / x]ph -> A.x[z / x]ph)
8		visset 2295	. . . . . . 7 |- z e. _V
9	8	sucex 3892	. . . . . 6 |- suc z e. _V
10	9	hbsbc1v 2464	. . . . 5 |- ([suc z / x]ph -> A.x[suc z / x]ph)
11	7, 10	hbim 1354	. . . 4 |- (([z / x]ph -> [suc z / x]ph) -> A.x([z / x]ph -> [suc z / x]ph))
12	6, 11	hbim 1354	. . 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 1957	. . . 4 |- (x = z -> (x e. On <-> z e. On))
14		sbequ12 1545	. . . . 5 |- (x = z -> (ph <-> [z / x]ph))
15		suceq 3729	. . . . . 6 |- (x = z -> suc x = suc z)
16		dfsbcq 2455	. . . . . 6 |- (suc x = suc z -> ([suc x / x]ph <-> [suc z / x]ph))
17	15, 16	syl 12	. . . . 5 |- (x = z -> ([suc x / x]ph <-> [suc z / x]ph))
18	14, 17	imbi12d 688	. . . 4 |- (x = z -> ((ph -> [suc x / x]ph) <-> ([z / x]ph -> [suc z / x]ph)))
19	13, 18	imbi12d 688	. . 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 1530	. 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 1317	. . . 4 |- (ph -> A.zph)
24	23, 7, 14	cbvral 2278	. . 3 |- (A.x e. y ph <-> A.z e. y [z / x]ph)
25	22, 24	syl5ibr 224	. 2 |- (Lim y -> (A.z e. y [z / x]ph -> [y / x]ph))
26	1, 2, 3, 4, 5, 21, 25	tfinds 3942	1 |- (x e. On -> ph)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  (/)c0 2875  Oncon0 3657  Lim wlim 3658  suc csuc 3659

This theorem is referenced by:  tfinds2 3947

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663

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