Theorem tfinds2 3947

Description: Transfinite Induction (inference schema), using implicit substititions. The first three hypotheses establish the substitutions we need. The last three are the basis and the induction hypotheses (for successor and limit ordinals respectively). Theorem Schema 4 of [Suppes] p. 197. The wff ta is an auxiliary antecedent to help shorten proofs using this theorem.

Hypotheses

Ref	Expression
tfinds2.1	|- (x = (/) -> (ph <-> ps))
tfinds2.2	|- (x = y -> (ph <-> ch))
tfinds2.3	|- (x = suc y -> (ph <-> th))
tfinds2.4	|- (ta -> ps)
tfinds2.5	|- (y e. On -> (ta -> (ch -> th)))
tfinds2.6	|- (Lim x -> (ta -> (A.y e. x ch -> ph)))

Assertion

Ref	Expression
tfinds2	|- (x e. On -> (ta -> ph))

Distinct variable groups:   x,y,ta   ps,x   ch,x   th,x   ph,y

Proof of Theorem tfinds2

Step	Hyp	Ref	Expression
1		tfinds2.4	. . 3 |- (ta -> ps)
2		0ex 3446	. . . 4 |- (/) e. _V
3		tfinds2.1	. . . . 5 |- (x = (/) -> (ph <-> ps))
4	3	imbi2d 674	. . . 4 |- (x = (/) -> ((ta -> ph) <-> (ta -> ps)))
5	2, 4	sbcie 2485	. . 3 |- ([(/) / x](ta -> ph) <-> (ta -> ps))
6	1, 5	mpbir 207	. 2 |- [(/) / x](ta -> ph)
7		tfinds2.5	. . . . . 6 |- (y e. On -> (ta -> (ch -> th)))
8	7	a2d 16	. . . . 5 |- (y e. On -> ((ta -> ch) -> (ta -> th)))
9	8	sbimi 1537	. . . 4 |- ([x / y]y e. On -> [x / y]((ta -> ch) -> (ta -> th)))
10		visset 2295	. . . . 5 |- x e. _V
11		sbcel1gv 2510	. . . . 5 |- (x e. _V -> ([x / y]y e. On <-> x e. On))
12	10, 11	ax-mp 7	. . . 4 |- ([x / y]y e. On <-> x e. On)
13		sbim 1604	. . . 4 |- ([x / y]((ta -> ch) -> (ta -> th)) <-> ([x / y](ta -> ch) -> [x / y](ta -> th)))
14	9, 12, 13	3imtr3i 235	. . 3 |- (x e. On -> ([x / y](ta -> ch) -> [x / y](ta -> th)))
15		tfinds2.2	. . . . . . 7 |- (x = y -> (ph <-> ch))
16	15	bicomd 580	. . . . . 6 |- (x = y -> (ch <-> ph))
17	16	equcoms 1489	. . . . 5 |- (y = x -> (ch <-> ph))
18	17	imbi2d 674	. . . 4 |- (y = x -> ((ta -> ch) <-> (ta -> ph)))
19	10, 18	sbcie 2485	. . 3 |- ([x / y](ta -> ch) <-> (ta -> ph))
20		visset 2295	. . . . . . 7 |- y e. _V
21	20	sucex 3892	. . . . . 6 |- suc y e. _V
22		tfinds2.3	. . . . . . 7 |- (x = suc y -> (ph <-> th))
23	22	imbi2d 674	. . . . . 6 |- (x = suc y -> ((ta -> ph) <-> (ta -> th)))
24	21, 23	sbcie 2485	. . . . 5 |- ([suc y / x](ta -> ph) <-> (ta -> th))
25	24	sbbii 1538	. . . 4 |- ([x / y][suc y / x](ta -> ph) <-> [x / y](ta -> th))
26		suceq 3729	. . . . 5 |- (x = y -> suc x = suc y)
27	26	sbcco2 2468	. . . 4 |- ([x / y][suc y / x](ta -> ph) <-> [suc x / x](ta -> ph))
28	25, 27	bitr3i 192	. . 3 |- ([x / y](ta -> th) <-> [suc x / x](ta -> ph))
29	14, 19, 28	3imtr3g 611	. 2 |- (x e. On -> ((ta -> ph) -> [suc x / x](ta -> ph)))
30		tfinds2.6	. . . . . . 7 |- (Lim x -> (ta -> (A.y e. x ch -> ph)))
31	30	a2d 16	. . . . . 6 |- (Lim x -> ((ta -> A.y e. x ch) -> (ta -> ph)))
32		r19.21v 2178	. . . . . 6 |- (A.y e. x (ta -> ch) <-> (ta -> A.y e. x ch))
33	31, 32	syl5ib 223	. . . . 5 |- (Lim x -> (A.y e. x (ta -> ch) -> (ta -> ph)))
34	33	sbimi 1537	. . . 4 |- ([y / x]Lim x -> [y / x](A.y e. x (ta -> ch) -> (ta -> ph)))
35		ax-17 1317	. . . . 5 |- (Lim y -> A.xLim y)
36		limeq 3669	. . . . 5 |- (x = y -> (Lim x <-> Lim y))
37	35, 36	sbie 1565	. . . 4 |- ([y / x]Lim x <-> Lim y)
38		sbim 1604	. . . 4 |- ([y / x](A.y e. x (ta -> ch) -> (ta -> ph)) <-> ([y / x]A.y e. x (ta -> ch) -> [y / x](ta -> ph)))
39	34, 37, 38	3imtr3i 235	. . 3 |- (Lim y -> ([y / x]A.y e. x (ta -> ch) -> [y / x](ta -> ph)))
40	18	sbralie 2453	. . 3 |- ([y / x]A.y e. x (ta -> ch) <-> A.x e. y (ta -> ph))
41	39, 40	syl5ibr 224	. 2 |- (Lim y -> (A.x e. y (ta -> ph) -> [y / x](ta -> ph)))
42	6, 29, 41	tfindes 3946	1 |- (x e. On -> (ta -> ph))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  _Vcvv 2292  (/)c0 2875  Oncon0 3657  Lim wlim 3658  suc csuc 3659

This theorem is referenced by:  abianfplem 5170

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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