Theorem tfinds2 3171

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 2716	. . . 4 |- (/) e. V
3		tfinds2.1	. . . . 5 |- (x = (/) -> (ph <-> ps))
4	3	imbi2d 614	. . . 4 |- (x = (/) -> ((ta -> ph) <-> (ta -> ps)))
5	2, 4	sbcie 1965	. . 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 1175	. . . 4 |- ([x / y]y e. On -> [x / y]((ta -> ch) -> (ta -> th)))
10		visset 1816	. . . . 5 |- x e. V
11		sbcel1gv 1983	. . . . 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 1236	. . . 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 523	. . . . . 6 |- (x = y -> (ch <-> ph))
17	16	equcoms 1132	. . . . 5 |- (y = x -> (ch <-> ph))
18	17	imbi2d 614	. . . 4 |- (y = x -> ((ta -> ch) <-> (ta -> ph)))
19	10, 18	sbcie 1965	. . 3 |- ([x / y](ta -> ch) <-> (ta -> ph))
20		visset 1816	. . . . . . 7 |- y e. V
21	20	sucex 3056	. . . . . 6 |- suc y e. V
22		tfinds2.3	. . . . . . 7 |- (x = suc y -> (ph <-> th))
23	22	imbi2d 614	. . . . . 6 |- (x = suc y -> ((ta -> ph) <-> (ta -> th)))
24	21, 23	sbcie 1965	. . . . 5 |- ([suc y / x](ta -> ph) <-> (ta -> th))
25	24	sbbii 1176	. . . 4 |- ([x / y][suc y / x](ta -> ph) <-> [x / y](ta -> th))
26		suceq 3040	. . . . 5 |- (x = y -> suc x = suc y)
27	26	sbcco2 1956	. . . 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 554	. 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 1719	. . . . . 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 1175	. . . 4 |- ([y / x]Lim x -> [y / x](A.y e. x (ta -> ch) -> (ta -> ph)))
35		ax-17 973	. . . . 5 |- (Lim y -> A.xLim y)
36		limeq 2966	. . . . 5 |- (x = y -> (Lim x <-> Lim y))
37	35, 36	sbie 1198	. . . 4 |- ([y / x]Lim x <-> Lim y)
38		sbim 1236	. . . 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 1944	. . 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 3170	1 |- (x e. On -> (ta -> ph))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  [wsbc 1172  A.wral 1648  Vcvv 1814  (/)c0 2283  Oncon0 2954  Lim wlim 2955  suc csuc 2956

This theorem is referenced by:  abianfplem 3967

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960

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