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Theorem tfinds2 6698
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:   , ,   ,   ,   ,   ,

Proof of Theorem tfinds2
StepHypRef Expression
1 tfinds2.4 . . 3
2 0ex 4582 . . . 4
3 tfinds2.1 . . . . 5
43imbi2d 316 . . . 4
52, 4sbcie 3362 . . 3
61, 5mpbir 209 . 2
7 vex 3112 . . . . . 6
8 tfinds2.5 . . . . . . . 8
98a2d 26 . . . . . . 7
109sbcth 3342 . . . . . 6
117, 10ax-mp 5 . . . . 5
12 sbcimg 3369 . . . . . 6
137, 12ax-mp 5 . . . . 5
1411, 13mpbi 208 . . . 4
15 sbcel1v 3392 . . . 4
16 sbcimg 3369 . . . . 5
177, 16ax-mp 5 . . . 4
1814, 15, 173imtr3i 265 . . 3
19 tfinds2.2 . . . . . . 7
2019bicomd 201 . . . . . 6
2120equcoms 1795 . . . . 5
2221imbi2d 316 . . . 4
237, 22sbcie 3362 . . 3
24 vex 3112 . . . . . . 7
2524sucex 6646 . . . . . 6
26 tfinds2.3 . . . . . . 7
2726imbi2d 316 . . . . . 6
2825, 27sbcie 3362 . . . . 5
2928sbcbii 3387 . . . 4
30 suceq 4948 . . . . 5
3130sbcco2 3351 . . . 4
3229, 31bitr3i 251 . . 3
3318, 23, 323imtr3g 269 . 2
34 sbsbc 3331 . . . 4
3522sbralie 3097 . . . 4
3634, 35bitr3i 251 . . 3
37 r19.21v 2862 . . . . . . . 8
38 tfinds2.6 . . . . . . . . 9
3938a2d 26 . . . . . . . 8
4037, 39syl5bi 217 . . . . . . 7
4140sbcth 3342 . . . . . 6
4224, 41ax-mp 5 . . . . 5
43 sbcimg 3369 . . . . . 6
4424, 43ax-mp 5 . . . . 5
4542, 44mpbi 208 . . . 4
46 limeq 4895 . . . . 5
4724, 46sbcie 3362 . . . 4
48 sbcimg 3369 . . . . 5
4924, 48ax-mp 5 . . . 4
5045, 47, 493imtr3i 265 . . 3
5136, 50syl5bir 218 . 2
526, 33, 51tfindes 6697 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  [wsb 1739  e.wcel 1818  A.wral 2807   cvv 3109  [.wsbc 3327   c0 3784   con0 4883  Limwlim 4884  succsuc 4885
This theorem is referenced by:  inar1  9174  grur1a  9218
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
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