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Theorem finds2 6728
Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)
Hypotheses
Ref Expression
finds2.1
finds2.2
finds2.3
finds2.4
finds2.5
Assertion
Ref Expression
finds2
Distinct variable groups:   , ,   ,   ,   ,   ,

Proof of Theorem finds2
StepHypRef Expression
1 finds2.4 . . . . 5
2 0ex 4582 . . . . . 6
3 finds2.1 . . . . . . 7
43imbi2d 316 . . . . . 6
52, 4elab 3246 . . . . 5
61, 5mpbir 209 . . . 4
7 finds2.5 . . . . . . 7
87a2d 26 . . . . . 6
9 vex 3112 . . . . . . 7
10 finds2.2 . . . . . . . 8
1110imbi2d 316 . . . . . . 7
129, 11elab 3246 . . . . . 6
139sucex 6646 . . . . . . 7
14 finds2.3 . . . . . . . 8
1514imbi2d 316 . . . . . . 7
1613, 15elab 3246 . . . . . 6
178, 12, 163imtr4g 270 . . . . 5
1817rgen 2817 . . . 4
19 peano5 6723 . . . 4
206, 18, 19mp2an 672 . . 3
2120sseli 3499 . 2
22 abid 2444 . 2
2321, 22sylib 196 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  e.wcel 1818  {cab 2442  A.wral 2807  C_wss 3475   c0 3784  succsuc 4885   com 6700
This theorem is referenced by:  finds1  6729  onnseq  7034  nnacl  7279  nnmcl  7280  nnecl  7281  nnacom  7285  nnaass  7290  nndi  7291  nnmass  7292  nnmsucr  7293  nnmcom  7294  nnmordi  7299  omsmolem  7321  isinf  7753  unblem2  7793  fiint  7817  dffi3  7911  card2inf  8002  cantnfle  8111  cantnflt  8112  cantnflem1  8129  cantnfleOLD  8141  cantnfltOLD  8142  cantnflem1OLD  8152  cnfcom  8165  cnfcomOLD  8173  trcl  8180  fseqenlem1  8426  infpssrlem3  8706  fin23lem26  8726  axdc3lem2  8852  axdc4lem  8856  axdclem2  8921  wunr1om  9118  wuncval2  9146  tskr1om  9166  grothomex  9228  peano5nni  10564  neibastop2lem  30178
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  df-om 6701
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