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Theorem finds 6726
 Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.)
Hypotheses
Ref Expression
finds.1
finds.2
finds.3
finds.4
finds.5
finds.6
Assertion
Ref Expression
finds
Distinct variable groups:   ,   ,   ,   ,   ,   ,   ,

Proof of Theorem finds
StepHypRef Expression
1 finds.5 . . . . 5
2 0ex 4582 . . . . . 6
3 finds.1 . . . . . 6
42, 3elab 3246 . . . . 5
51, 4mpbir 209 . . . 4
6 finds.6 . . . . . 6
7 vex 3112 . . . . . . 7
8 finds.2 . . . . . . 7
97, 8elab 3246 . . . . . 6
107sucex 6646 . . . . . . 7
11 finds.3 . . . . . . 7
1210, 11elab 3246 . . . . . 6
136, 9, 123imtr4g 270 . . . . 5
1413rgen 2817 . . . 4
15 peano5 6723 . . . 4
165, 14, 15mp2an 672 . . 3
1716sseli 3499 . 2
18 finds.4 . . 3
1918elabg 3247 . 2
2017, 19mpbid 210 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:  findsg  6727  findes  6730  seqomlem1  7134  nna0r  7277  nnm0r  7278  nnawordi  7289  nneob  7320  nneneq  7720  pssnn  7758  inf3lem1  8066  inf3lem2  8067  cantnfval2  8109  cantnfp1lem3  8120  cantnfval2OLD  8139  cantnfp1lem3OLD  8146  r1fin  8212  ackbij1lem14  8634  ackbij1lem16  8636  ackbij1  8639  ackbij2lem2  8641  ackbij2lem3  8642  infpssrlem4  8707  fin23lem14  8734  fin23lem34  8747  itunitc1  8821  ituniiun  8823  om2uzuzi  12060  om2uzlti  12061  om2uzrdg  12067  uzrdgxfr  12077  hashgadd  12445  mreexexd  15045  trpredmintr  29314  findfvcl  29917 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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