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Theorem finds1 6729
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, 22-Mar-2006.)
Hypotheses
Ref Expression
finds1.1
finds1.2
finds1.3
finds1.4
finds1.5
Assertion
Ref Expression
finds1
Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem finds1
StepHypRef Expression
1 eqid 2457 . 2
2 finds1.1 . . 3
3 finds1.2 . . 3
4 finds1.3 . . 3
5 finds1.4 . . . 4
65a1i 11 . . 3
7 finds1.5 . . . 4
87a1d 25 . . 3
92, 3, 4, 6, 8finds2 6728 . 2
101, 9mpi 17 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  e.wcel 1818   c0 3784  succsuc 4885   com 6700
This theorem is referenced by:  findcard  7779  findcard2  7780  pwfi  7835  alephfplem3  8508  pwsdompw  8605  hsmexlem4  8830
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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