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Theorem findes 4702
 Description: Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. See tfindes 4669 for the transfinite version. (Contributed by Raph Levien, 9-Jul-2003.)
Hypotheses
Ref Expression
findes.1
findes.2
Assertion
Ref Expression
findes

Proof of Theorem findes
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3007 . 2
2 sbequ 2013 . 2
3 dfsbcq2 3007 . 2
4 sbequ12r 1873 . 2
5 findes.1 . 2
6 nfv 1609 . . . 4
7 nfs1v 2058 . . . . 5
8 nfsbc1v 3023 . . . . 5
97, 8nfim 1781 . . . 4
106, 9nfim 1781 . . 3
11 eleq1 2356 . . . 4
12 sbequ12 1872 . . . . 5
13 suceq 4473 . . . . . 6
14 dfsbcq 3006 . . . . . 6
1513, 14syl 15 . . . . 5
1612, 15imbi12d 311 . . . 4
1711, 16imbi12d 311 . . 3
18 findes.2 . . 3
1910, 17, 18chvar 1939 . 2
201, 2, 3, 4, 5, 19finds 4698 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wceq 1632  wsb 1638   wcel 1696  wsbc 3004  c0 3468   csuc 4410  com 4672 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673
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