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Theorem findes 4561
 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
Assertion
Ref Expression
findes

Proof of Theorem findes
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2940 . 2
2 sbequ 1820 . 2
3 dfsbcq2 2940 . 2
4 sbequ12r 1752 . 2
5 findes.1 . 2
6 nfv 1508 . . . 4
7 nfs1v 1919 . . . . 5
8 nfsbc1v 2955 . . . . 5
97, 8nfim 1552 . . . 4
106, 9nfim 1552 . . 3
11 eleq1 2220 . . . 4
12 sbequ12 1751 . . . . 5
13 suceq 4362 . . . . . 6
14 dfsbcq 2939 . . . . . 6
1513, 14syl 14 . . . . 5
1612, 15imbi12d 233 . . . 4
1711, 16imbi12d 233 . . 3
18 findes.2 . . 3
1910, 17, 18chvar 1737 . 2
201, 2, 3, 4, 5, 19finds 4558 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104   wceq 1335  wsb 1742   wcel 2128  wsbc 2937  c0 3394   csuc 4325  com 4548 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-iinf 4546 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-suc 4331  df-iom 4549 This theorem is referenced by: (None)
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