Theorem tfinds2 3155

Description: Transfinite Induction (inference schema) with implicit substitutions. 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 2701	. . . 4 |- (/) e. V
3		tfinds2.1	. . . . 5 |- (x = (/) -> (ph <-> ps))
4	3	imbi2d 610	. . . 4 |- (x = (/) -> ((ta -> ph) <-> (ta -> ps)))
5	2, 4	sbcie 1952	. . 3 |- ([(/) / x](ta -> ph) <-> (ta -> ps))
6	1, 5	mpbir 190	. 2 |- [(/) / x](ta -> ph)
7		tfinds2.5	. . . . . 6 |- (y e. On -> (ta -> (ch -> th)))
8	7	a2d 13	. . . . 5 |- (y e. On -> ((ta -> ch) -> (ta -> th)))
9	8	sbimi 1169	. . . 4 |- ([x / y]y e. On -> [x / y]((ta -> ch) -> (ta -> th)))
10		visset 1804	. . . . 5 |- x e. V
11		sbcel1gv 1970	. . . . 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 1229	. . . 4 |- ([x / y]((ta -> ch) -> (ta -> th)) <-> ([x / y](ta -> ch) -> [x / y](ta -> th)))
14	9, 12, 13	3imtr3 218	. . 3 |- (x e. On -> ([x / y](ta -> ch) -> [x / y](ta -> th)))
15		tfinds2.2	. . . . . . 7 |- (x = y -> (ph <-> ch))
16	15	bicomd 519	. . . . . 6 |- (x = y -> (ch <-> ph))
17	16	equcoms 1126	. . . . 5 |- (y = x -> (ch <-> ph))
18	17	imbi2d 610	. . . 4 |- (y = x -> ((ta -> ch) <-> (ta -> ph)))
19	10, 18	sbcie 1952	. . 3 |- ([x / y](ta -> ch) <-> (ta -> ph))
20		visset 1804	. . . . . . 7 |- y e. V
21	20	sucex 3040	. . . . . 6 |- suc y e. V
22		tfinds2.3	. . . . . . 7 |- (x = suc y -> (ph <-> th))
23	22	imbi2d 610	. . . . . 6 |- (x = suc y -> ((ta -> ph) <-> (ta -> th)))
24	21, 23	sbcie 1952	. . . . 5 |- ([suc y / x](ta -> ph) <-> (ta -> th))
25	24	sbbii 1170	. . . 4 |- ([x / y][suc y / x](ta -> ph) <-> [x / y](ta -> th))
26		suceq 3024	. . . . 5 |- (x = y -> suc x = suc y)
27	26	sbcco2 1943	. . . 4 |- ([x / y][suc y / x](ta -> ph) <-> [suc x / x](ta -> ph))
28	25, 27	bitr3 175	. . 3 |- ([x / y](ta -> th) <-> [suc x / x](ta -> ph))
29	14, 19, 28	3imtr3g 550	. 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 13	. . . . . 6 |- (Lim x -> ((ta -> A.y e. x ch) -> (ta -> ph)))
32		r19.21v 1708	. . . . . 6 |- (A.y e. x (ta -> ch) <-> (ta -> A.y e. x ch))
33	31, 32	syl5ib 206	. . . . 5 |- (Lim x -> (A.y e. x (ta -> ch) -> (ta -> ph)))
34	33	sbimi 1169	. . . 4 |- ([y / x]Lim x -> [y / x](A.y e. x (ta -> ch) -> (ta -> ph)))
35		ax-17 968	. . . . 5 |- (Lim y -> A.xLim y)
36		limeq 2950	. . . . 5 |- (x = y -> (Lim x <-> Lim y))
37	35, 36	sbie 1192	. . . 4 |- ([y / x]Lim x <-> Lim y)
38		sbim 1229	. . . 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	3imtr3 218	. . 3 |- (Lim y -> ([y / x]A.y e. x (ta -> ch) -> [y / x](ta -> ph)))
40	18	sbralie 1931	. . 3 |- ([y / x]A.y e. x (ta -> ch) <-> A.x e. y (ta -> ph))
41	39, 40	syl5ibr 207	. 2 |- (Lim y -> (A.x e. y (ta -> ph) -> [y / x](ta -> ph)))
42	6, 29, 41	tfindes 3154	1 |- (x e. On -> (ta -> ph))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955  [wsbc 1166  A.wral 1637  Vcvv 1802  (/)c0 2270  Oncon0 2938  Lim wlim 2939  suc csuc 2940

This theorem is referenced by:  abianfplem 3946

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944

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