Theorem tfinds3 6699

Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.)

Hypotheses

Ref	Expression
tfinds3.1
tfinds3.2
tfinds3.3
tfinds3.4
tfinds3.5
tfinds3.6
tfinds3.7

Assertion

Ref	Expression
tfinds3

Distinct variable groups:   ,   ,   ,   ,   ,,

Proof of Theorem tfinds3

Step	Hyp	Ref	Expression
1		tfinds3.1	. . 3
2	1	imbi2d 316	. 2
3		tfinds3.2	. . 3
4	3	imbi2d 316	. 2
5		tfinds3.3	. . 3
6	5	imbi2d 316	. 2
7		tfinds3.4	. . 3
8	7	imbi2d 316	. 2
9		tfinds3.5	. 2
10		tfinds3.6	. . 3
11	10	a2d 26	. 2
12		r19.21v 2862	. . 3
13		tfinds3.7	. . . 4
14	13	a2d 26	. . 3
15	12, 14	syl5bi 217	. 2
16	2, 4, 6, 8, 9, 11, 15	tfinds 6694	1

Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  e.wcel 1818  A.wral 2807  c0 3784  con0 4883  Limwlim 4884  succsuc 4885

This theorem is referenced by:  oacl  7204  omcl  7205  oecl  7206  oawordri  7218  oaass  7229  oarec  7230  omordi  7234  omwordri  7240  odi  7247  omass  7248  oen0  7254  oewordri  7260  oeworde  7261  oeoelem  7266  omabs  7315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889