Theorem findes 6501

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 6468 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	|- [. (/) / x ]. ph
findes.2	|- ( x e. om -> ( ph -> [. suc x / x ]. ph ) )

Assertion

Ref	Expression
findes	|- ( x e. om -> ph )

Proof of Theorem findes

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3184	. 2 |- ( z = (/) -> ( [ z / x ] ph <-> [. (/) / x ]. ph ) )
2		sbequ 2067	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
3		dfsbcq2 3184	. 2 |- ( z = suc y -> ( [ z / x ] ph <-> [. suc y / x ]. ph ) )
4		sbequ12r 1937	. 2 |- ( z = x -> ( [ z / x ] ph <-> ph ) )
5		findes.1	. 2 |- [. (/) / x ]. ph
6		nfv 1673	. . . 4 |- F/ x y e. om
7		nfs1v 2142	. . . . 5 |- F/ x [ y / x ] ph
8		nfsbc1v 3201	. . . . 5 |- F/ x [. suc y / x ]. ph
9	7, 8	nfim 1852	. . . 4 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
10	6, 9	nfim 1852	. . 3 |- F/ x ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) )
11		eleq1 2498	. . . 4 |- ( x = y -> ( x e. om <-> y e. om ) )
12		sbequ12 1936	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
13		suceq 4779	. . . . . 6 |- ( x = y -> suc x = suc y )
14		dfsbcq 3183	. . . . . 6 |- ( suc x = suc y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) )
15	13, 14	syl 16	. . . . 5 |- ( x = y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) )
16	12, 15	imbi12d 320	. . . 4 |- ( x = y -> ( ( ph -> [. suc x / x ]. ph ) <-> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) )
17	11, 16	imbi12d 320	. . 3 |- ( x = y -> ( ( x e. om -> ( ph -> [. suc x / x ]. ph ) ) <-> ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) ) )
18		findes.2	. . 3 |- ( x e. om -> ( ph -> [. suc x / x ]. ph ) )
19	10, 17, 18	chvar 1957	. 2 |- ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) )
20	1, 2, 3, 4, 5, 19	finds 6497	1 |- ( x e. om -> ph )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1369   [wsb 1700   e. wcel 1756   [.wsbc 3181   (/)c0 3632   suc csuc 4716   omcom 6471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-tr 4381  df-eprel 4627  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-om 6472

This theorem is referenced by: (None)