Theorem tfinds2 2405

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 group(s):   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 1745	. . . 4 |- (/) e. V
3		tfinds2.1	. . . . 5 |- (x = (/) -> (ph <-> ps))
4	3	imbi2d 464	. . . 4 |- (x = (/) -> ((ta -> ph) <-> (ta -> ps)))
5	2, 4	sbcie 1455	. . 3 |- ([(/) / x](ta -> ph) <-> (ta -> ps))
6	1, 5	mpbir 165	. 2 |- [(/) / x](ta -> ph)
7		tfinds2.5	. . . . . 6 |- (y e. On -> (ta -> (ch -> th)))
8	7	a2d 15	. . . . 5 |- (y e. On -> ((ta -> ch) -> (ta -> th)))
9	8	sbimi 854	. . . 4 |- ([x / y]y e. On -> [x / y]((ta -> ch) -> (ta -> th)))
10		visset 1350	. . . . 5 |- x e. V
11		sbcel1 1466	. . . . 5 |- (x e. V -> ([x / y]y e. On <-> x e. On))
12	10, 11	ax-mp 6	. . . 4 |- ([x / y]y e. On <-> x e. On)
13		sbim 886	. . . 4 |- ([x / y]((ta -> ch) -> (ta -> th)) <-> ([x / y](ta -> ch) -> [x / y](ta -> th)))
14	9, 12, 13	3imtr3 191	. . 3 |- (x e. On -> ([x / y](ta -> ch) -> [x / y](ta -> th)))
15		tfinds2.2	. . . . . . 7 |- (x = y -> (ph <-> ch))
16	15	bicomd 399	. . . . . 6 |- (x = y -> (ch <-> ph))
17	16	eqcoms 813	. . . . 5 |- (y = x -> (ch <-> ph))
18	17	imbi2d 464	. . . 4 |- (y = x -> ((ta -> ch) <-> (ta -> ph)))
19	10, 18	sbcie 1455	. . 3 |- ([x / y](ta -> ch) <-> (ta -> ph))
20		visset 1350	. . . . . . 7 |- y e. V
21	20	sucex 2303	. . . . . 6 |- suc y e. V
22		tfinds2.3	. . . . . . 7 |- (x = suc y -> (ph <-> th))
23	22	imbi2d 464	. . . . . 6 |- (x = suc y -> ((ta -> ph) <-> (ta -> th)))
24	21, 23	sbcie 1455	. . . . 5 |- ([suc y / x](ta -> ph) <-> (ta -> th))
25	24	bisb 855	. . . 4 |- ([x / y][suc y / x](ta -> ph) <-> [x / y](ta -> th))
26		suceq 2288	. . . . 5 |- (x = y -> suc x = suc y)
27	26	sbcco2 1449	. . . 4 |- ([x / y][suc y / x](ta -> ph) <-> [suc x / x](ta -> ph))
28	25, 27	bitr3 153	. . 3 |- ([x / y](ta -> th) <-> [suc x / x](ta -> ph))
29	14, 19, 28	3imtr3g 425	. 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 15	. . . . . 6 |- (Lim x -> ((ta -> A.y e. x ch) -> (ta -> ph)))
32		r19.21v 1260	. . . . . 6 |- (A.y e. x (ta -> ch) <-> (ta -> A.y e. x ch))
33	31, 32	syl5ib 181	. . . . 5 |- (Lim x -> (A.y e. x (ta -> ch) -> (ta -> ph)))
34	33	sbimi 854	. . . 4 |- ([y / x]Lim x -> [y / x](A.y e. x (ta -> ch) -> (ta -> ph)))
35		ax-17 925	. . . . 5 |- (Lim y -> A.xLim y)
36		limeq 2211	. . . . 5 |- (x = y -> (Lim x <-> Lim y))
37	35, 36	sbie 904	. . . 4 |- ([y / x]Lim x <-> Lim y)
38		sbim 886	. . . 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 191	. . 3 |- (Lim y -> ([y / x]A.y e. x (ta -> ch) -> [y / x](ta -> ph)))
40	18	sbralie 1439	. . 3 |- ([y / x]A.y e. x (ta -> ch) <-> A.x e. y (ta -> ph))
41	39, 40	syl5ibr 182	. 2 |- (Lim y -> (A.x e. y (ta -> ph) -> [y / x](ta -> ph)))
42	6, 29, 41	tfindes 2404	1 |- (x e. On -> (ta -> ph))

Syntax hints:   -> wi 2   <-> wb 127   = weq 797  [wsb 852   = wceq 1091   e. wcel 1092  A.wral 1201  Vcvv 1348  [wsbc 1440  (/)c0 1707  Oncon0 2199  Lim wlim 2200  suc csuc 2201

This theorem is referenced by:  abianfplem 2999

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205

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