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Theorem findes 4253
Description: Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.)
Hypotheses
Ref Expression
findes.1  [. (/)  ].
findes.2  om  [. suc  ].
Assertion
Ref Expression
findes  om

Proof of Theorem findes
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2744 . 2  (/)  [. (/)  ].
2 sbequ 1703 . 2
3 dfsbcq2 2744 . 2  suc  [. suc  ].
4 sbequ12r 1637 . 2
5 findes.1 . 2  [. (/)  ].
6 nfv 1402 . . . 4  F/  om
7 nfs1v 1797 . . . . 5  F/
8 nfsbc1v 2759 . . . . 5  F/ [.
suc  ].
97, 8nfim 1446 . . . 4  F/  [. suc  ].
106, 9nfim 1446 . . 3  F/  om  [. suc  ].
11 eleq1 2082 . . . 4  om 
om
12 sbequ12 1636 . . . . 5
13 suceq 4088 . . . . . 6  suc  suc
14 dfsbcq 2743 . . . . . 6  suc  suc  [. suc  ].  [. suc  ].
1513, 14syl 14 . . . . 5  [. suc  ]. 
[. suc  ].
1612, 15imbi12d 223 . . . 4  [. suc  ].  [. suc  ].
1711, 16imbi12d 223 . . 3  om  [. suc  ].  om  [. suc  ].
18 findes.2 . . 3  om  [. suc  ].
1910, 17, 18chvar 1622 . 2  om  [. suc  ].
201, 2, 3, 4, 5, 19finds 4250 1  om
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wceq 1228   wcel 1374  wsb 1627   [.wsbc 2741   (/)c0 3201   suc csuc 4051   omcom 4240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241
This theorem is referenced by: (None)
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