Theorem tfinds2 4835

Description: Transfinite Induction (inference schema), using 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 is an auxiliary antecedent to help shorten proofs using this theorem. (Contributed by NM, 4-Sep-2004.)

Hypotheses

Ref	Expression
tfinds2.1
tfinds2.2
tfinds2.3
tfinds2.4
tfinds2.5
tfinds2.6

Assertion

Ref	Expression
tfinds2

Distinct variable groups:   ,,   ,   ,   ,   ,

Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem tfinds2

Step	Hyp	Ref	Expression
1		tfinds2.4	. . 3
2		0ex 4331	. . . 4
3		tfinds2.1	. . . . 5
4	3	imbi2d 308	. . . 4
5	2, 4	sbcie 3187	. . 3
6	1, 5	mpbir 201	. 2
7		vex 2951	. . . . . 6
8		tfinds2.5	. . . . . . . 8
9	8	a2d 24	. . . . . . 7
10	9	sbcth 3167	. . . . . 6
11	7, 10	ax-mp 8	. . . . 5
12		sbcimg 3194	. . . . . 6
13	7, 12	ax-mp 8	. . . . 5
14	11, 13	mpbi 200	. . . 4
15		sbcel1gv 3212	. . . . 5
16	7, 15	ax-mp 8	. . . 4
17		sbcimg 3194	. . . . 5
18	7, 17	ax-mp 8	. . . 4
19	14, 16, 18	3imtr3i 257	. . 3
20		tfinds2.2	. . . . . . 7
21	20	bicomd 193	. . . . . 6
22	21	equcoms 1693	. . . . 5
23	22	imbi2d 308	. . . 4
24	7, 23	sbcie 3187	. . 3
25		vex 2951	. . . . . . 7
26	25	sucex 4783	. . . . . 6
27		tfinds2.3	. . . . . . 7
28	27	imbi2d 308	. . . . . 6
29	26, 28	sbcie 3187	. . . . 5
30	29	sbcbii 3208	. . . 4
31		suceq 4638	. . . . 5
32	31	sbcco2 3176	. . . 4
33	30, 32	bitr3i 243	. . 3
34	19, 24, 33	3imtr3g 261	. 2
35		sbsbc 3157	. . . 4
36	23	sbralie 2937	. . . 4
37	35, 36	bitr3i 243	. . 3
38		r19.21v 2785	. . . . . . . 8
39		tfinds2.6	. . . . . . . . 9
40	39	a2d 24	. . . . . . . 8
41	38, 40	syl5bi 209	. . . . . . 7
42	41	sbcth 3167	. . . . . 6
43	25, 42	ax-mp 8	. . . . 5
44		sbcimg 3194	. . . . . 6
45	25, 44	ax-mp 8	. . . . 5
46	43, 45	mpbi 200	. . . 4
47		limeq 4585	. . . . 5
48	25, 47	sbcie 3187	. . . 4
49		sbcimg 3194	. . . . 5
50	25, 49	ax-mp 8	. . . 4
51	46, 48, 50	3imtr3i 257	. . 3
52	37, 51	syl5bir 210	. 2
53	6, 34, 52	tfindes 4834	1

Syntax hints:  ->wi 4  <->wb 177  =wceq 1652  [wsb 1658  e.wcel 1725  A.wral 2697  cvv 2948  [.wsbc 3153  c0 3620  con0 4573  Limwlim 4574  succsuc 4575

This theorem is referenced by:  abianfplem  6707  inar1  8642  grur1a  8686

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579