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Theorem tfindes 6697
Description: Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction step for successors, and the third is the induction step 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:   ,   ,

Proof of Theorem tfindes
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3329 . 2
2 dfsbcq 3329 . 2
3 dfsbcq 3329 . 2
4 sbceq2a 3339 . 2
5 tfindes.1 . 2
6 nfv 1707 . . . 4
7 nfsbc1v 3347 . . . . 5
8 nfsbc1v 3347 . . . . 5
97, 8nfim 1920 . . . 4
106, 9nfim 1920 . . 3
11 eleq1 2529 . . . 4
12 sbceq1a 3338 . . . . 5
13 suceq 4948 . . . . . 6
1413sbceq1d 3332 . . . . 5
1512, 14imbi12d 320 . . . 4
1611, 15imbi12d 320 . . 3
17 tfindes.2 . . 3
1810, 16, 17chvar 2013 . 2
19 cbvralsv 3095 . . . 4
20 sbsbc 3331 . . . . 5
2120ralbii 2888 . . . 4
2219, 21bitri 249 . . 3
23 tfindes.3 . . 3
2422, 23syl5bir 218 . 2
251, 2, 3, 4, 5, 18, 24tfinds 6694 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  [wsb 1739  e.wcel 1818  A.wral 2807  [.wsbc 3327   c0 3784   con0 4883  Limwlim 4884  succsuc 4885
This theorem is referenced by:  tfinds2  6698
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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