Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfindes Unicode version

Theorem tfindes 4653
 Description: Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction hypothesis for successors, and the third is the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 5-Mar-2004.)
Hypotheses
Ref Expression
tfindes.1
tfindes.2
tfindes.3
Assertion
Ref Expression
tfindes
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem tfindes
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2993 . 2
2 dfsbcq 2993 . 2
3 dfsbcq 2993 . 2
4 sbceq2a 3002 . 2
5 tfindes.1 . 2
6 nfv 1605 . . . 4
7 nfsbc1v 3010 . . . . 5
8 nfsbc1v 3010 . . . . 5
97, 8nfim 1769 . . . 4
106, 9nfim 1769 . . 3
11 eleq1 2343 . . . 4
12 sbceq1a 3001 . . . . 5
13 suceq 4457 . . . . . 6
14 dfsbcq 2993 . . . . . 6
1513, 14syl 15 . . . . 5
1612, 15imbi12d 311 . . . 4
1711, 16imbi12d 311 . . 3
18 tfindes.2 . . 3
1910, 17, 18chvar 1926 . 2
20 cbvralsv 2775 . . . 4
21 sbsbc 2995 . . . . 5
2221ralbii 2567 . . . 4
2320, 22bitri 240 . . 3
24 tfindes.3 . . 3
2523, 24syl5bir 209 . 2
261, 2, 3, 4, 5, 19, 25tfinds 4650 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wceq 1623  wsb 1629   wcel 1684  wral 2543  wsbc 2991  c0 3455  con0 4392   wlim 4393   csuc 4394 This theorem is referenced by:  tfinds2  4654 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398
 Copyright terms: Public domain W3C validator