Theorem tfindes 3159

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 1939	. 2 |- (y = (/) -> ([y / x]ph <-> [(/) / x]ph))
2		sbequ 1227	. 2 |- (y = z -> ([y / x]ph <-> [z / x]ph))
3		dfsbcq 1939	. 2 |- (y = suc z -> ([y / x]ph <-> [suc z / x]ph))
4		sbequ12r 1180	. 2 |- (y = x -> ([y / x]ph <-> ph))
5		tfindes.1	. 2 |- [(/) / x]ph
6		ax-17 969	. . . 4 |- (z e. On -> A.x z e. On)
7		hbs1 1330	. . . . 5 |- ([z / x]ph -> A.x[z / x]ph)
8		visset 1809	. . . . . . 7 |- z e. V
9	8	sucex 3045	. . . . . 6 |- suc z e. V
10	9	hbsbc1v 1946	. . . . 5 |- ([suc z / x]ph -> A.x[suc z / x]ph)
11	7, 10	hbim 1005	. . . 4 |- (([z / x]ph -> [suc z / x]ph) -> A.x([z / x]ph -> [suc z / x]ph))
12	6, 11	hbim 1005	. . 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 1531	. . . 4 |- (x = z -> (x e. On <-> z e. On))
14		sbequ12 1179	. . . . 5 |- (x = z -> (ph <-> [z / x]ph))
15		suceq 3029	. . . . . 6 |- (x = z -> suc x = suc z)
16		dfsbcq 1939	. . . . . 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 625	. . . 4 |- (x = z -> ((ph -> [suc x / x]ph) <-> ([z / x]ph -> [suc z / x]ph)))
19	13, 18	imbi12d 625	. . 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 1165	. 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-17 969	. . . 4 |- (ph -> A.zph)
24	23, 7, 14	cbvral 1794	. . 3 |- (A.x e. y ph <-> A.z e. y [z / x]ph)
25	22, 24	syl5ibr 207	. 2 |- (Lim y -> (A.z e. y [z / x]ph -> [y / x]ph))
26	1, 2, 3, 4, 5, 21, 25	tfinds 3156	1 |- (x e. On -> ph)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 954   e. wcel 956  [wsbc 1168  A.wral 1642  (/)c0 2276  Oncon0 2943  Lim wlim 2944  suc csuc 2945

This theorem is referenced by:  tfinds2 3160

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949

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