Theorem tfinds2 3165

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 2711	. . . 4 |- (/) e. V
3		tfinds2.1	. . . . 5 |- (x = (/) -> (ph <-> ps))
4	3	imbi2d 612	. . . 4 |- (x = (/) -> ((ta -> ph) <-> (ta -> ps)))
5	2, 4	sbcie 1962	. . 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 1173	. . . 4 |- ([x / y]y e. On -> [x / y]((ta -> ch) -> (ta -> th)))
10		visset 1813	. . . . 5 |- x e. V
11		sbcel1gv 1980	. . . . 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 1234	. . . 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 521	. . . . . 6 |- (x = y -> (ch <-> ph))
17	16	equcoms 1130	. . . . 5 |- (y = x -> (ch <-> ph))
18	17	imbi2d 612	. . . 4 |- (y = x -> ((ta -> ch) <-> (ta -> ph)))
19	10, 18	sbcie 1962	. . 3 |- ([x / y](ta -> ch) <-> (ta -> ph))
20		visset 1813	. . . . . . 7 |- y e. V
21	20	sucex 3050	. . . . . 6 |- suc y e. V
22		tfinds2.3	. . . . . . 7 |- (x = suc y -> (ph <-> th))
23	22	imbi2d 612	. . . . . 6 |- (x = suc y -> ((ta -> ph) <-> (ta -> th)))
24	21, 23	sbcie 1962	. . . . 5 |- ([suc y / x](ta -> ph) <-> (ta -> th))
25	24	sbbii 1174	. . . 4 |- ([x / y][suc y / x](ta -> ph) <-> [x / y](ta -> th))
26		suceq 3034	. . . . 5 |- (x = y -> suc x = suc y)
27	26	sbcco2 1953	. . . 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 552	. 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 1716	. . . . . 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 1173	. . . 4 |- ([y / x]Lim x -> [y / x](A.y e. x (ta -> ch) -> (ta -> ph)))
35		ax-17 971	. . . . 5 |- (Lim y -> A.xLim y)
36		limeq 2960	. . . . 5 |- (x = y -> (Lim x <-> Lim y))
37	35, 36	sbie 1196	. . . 4 |- ([y / x]Lim x <-> Lim y)
38		sbim 1234	. . . 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 1941	. . 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 3164	1 |- (x e. On -> (ta -> ph))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  [wsbc 1170  A.wral 1645  Vcvv 1811  (/)c0 2280  Oncon0 2948  Lim wlim 2949  suc csuc 2950

This theorem is referenced by:  abianfplem 3961

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954

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