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Theorem findes 6730
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. See tfindes 6697 for the transfinite version. 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 3330 . 2
2 sbequ 2117 . 2
3 dfsbcq2 3330 . 2
4 sbequ12r 1993 . 2
5 findes.1 . 2
6 nfv 1707 . . . 4
7 nfs1v 2181 . . . . 5
8 nfsbc1v 3347 . . . . 5
97, 8nfim 1920 . . . 4
106, 9nfim 1920 . . 3
11 eleq1 2529 . . . 4
12 sbequ12 1992 . . . . 5
13 suceq 4948 . . . . . 6
1413sbceq1d 3332 . . . . 5
1512, 14imbi12d 320 . . . 4
1611, 15imbi12d 320 . . 3
17 findes.2 . . 3
1810, 16, 17chvar 2013 . 2
191, 2, 3, 4, 5, 18finds 6726 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  [wsb 1739  e.wcel 1818  [.wsbc 3327   c0 3784  succsuc 4885   com 6700
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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