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Theorem tfinds2 4884
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
StepHypRef Expression
1 tfinds2.4 . . 3
2 0ex 4373 . . . 4
3 tfinds2.1 . . . . 5
43imbi2d 309 . . . 4
52, 4sbcie 3204 . . 3
61, 5mpbir 202 . 2
7 vex 2968 . . . . . 6
8 tfinds2.5 . . . . . . . 8
98a2d 25 . . . . . . 7
109sbcth 3184 . . . . . 6
117, 10ax-mp 5 . . . . 5
12 sbcimg 3211 . . . . . 6
137, 12ax-mp 5 . . . . 5
1411, 13mpbi 201 . . . 4
15 sbcel1v 3233 . . . 4
16 sbcimg 3211 . . . . 5
177, 16ax-mp 5 . . . 4
1814, 15, 173imtr3i 258 . . 3
19 tfinds2.2 . . . . . . 7
2019bicomd 194 . . . . . 6
2120equcoms 1696 . . . . 5
2221imbi2d 309 . . . 4
237, 22sbcie 3204 . . 3
24 vex 2968 . . . . . . 7
2524sucex 4832 . . . . . 6
26 tfinds2.3 . . . . . . 7
2726imbi2d 309 . . . . . 6
2825, 27sbcie 3204 . . . . 5
2928sbcbii 3228 . . . 4
30 suceq 4687 . . . . 5
3130sbcco2 3193 . . . 4
3229, 31bitr3i 244 . . 3
3318, 23, 323imtr3g 262 . 2
34 sbsbc 3174 . . . 4
3522sbralie 2954 . . . 4
3634, 35bitr3i 244 . . 3
37 r19.21v 2800 . . . . . . . 8
38 tfinds2.6 . . . . . . . . 9
3938a2d 25 . . . . . . . 8
4037, 39syl5bi 210 . . . . . . 7
4140sbcth 3184 . . . . . 6
4224, 41ax-mp 5 . . . . 5
43 sbcimg 3211 . . . . . 6
4424, 43ax-mp 5 . . . . 5
4542, 44mpbi 201 . . . 4
46 limeq 4634 . . . . 5
4724, 46sbcie 3204 . . . 4
48 sbcimg 3211 . . . . 5
4924, 48ax-mp 5 . . . 4
5045, 47, 493imtr3i 258 . . 3
5136, 50syl5bir 211 . 2
526, 33, 51tfindes 4883 1
Colors of variables: wff set class
Syntax hints:  ->wi 4  <->wb 178  =wceq 1654  [wsb 1660  e.wcel 1728  A.wral 2712   cvv 2965  [.wsbc 3170   c0 3616   con0 4622  Limwlim 4623  succsuc 4624
This theorem is referenced by:  abianfplem  6764  inar1  8701  grur1a  8745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4364  ax-nul 4372  ax-pr 4442  ax-un 4742
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3766  df-pw 3828  df-sn 3847  df-pr 3848  df-tp 3849  df-op 3850  df-uni 4044  df-br 4244  df-opab 4302  df-tr 4337  df-eprel 4535  df-po 4544  df-so 4545  df-fr 4582  df-we 4584  df-ord 4625  df-on 4626  df-lim 4627  df-suc 4628
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