Theorem findes 6715

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 6682 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 3316	. 2 |- ( z = (/) -> ( [ z / x ] ph <-> [. (/) / x ]. ph ) )
2		sbequ 2103	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
3		dfsbcq2 3316	. 2 |- ( z = suc y -> ( [ z / x ] ph <-> [. suc y / x ]. ph ) )
4		sbequ12r 1979	. 2 |- ( z = x -> ( [ z / x ] ph <-> ph ) )
5		findes.1	. 2 |- [. (/) / x ]. ph
6		nfv 1694	. . . 4 |- F/ x y e. om
7		nfs1v 2167	. . . . 5 |- F/ x [ y / x ] ph
8		nfsbc1v 3333	. . . . 5 |- F/ x [. suc y / x ]. ph
9	7, 8	nfim 1906	. . . 4 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
10	6, 9	nfim 1906	. . 3 |- F/ x ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) )
11		eleq1 2515	. . . 4 |- ( x = y -> ( x e. om <-> y e. om ) )
12		sbequ12 1978	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
13		suceq 4933	. . . . . 6 |- ( x = y -> suc x = suc y )
14	13	sbceq1d 3318	. . . . 5 |- ( x = y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) )
15	12, 14	imbi12d 320	. . . 4 |- ( x = y -> ( ( ph -> [. suc x / x ]. ph ) <-> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) )
16	11, 15	imbi12d 320	. . 3 |- ( x = y -> ( ( x e. om -> ( ph -> [. suc x / x ]. ph ) ) <-> ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) ) )
17		findes.2	. . 3 |- ( x e. om -> ( ph -> [. suc x / x ]. ph ) )
18	10, 16, 17	chvar 1999	. 2 |- ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) )
19	1, 2, 3, 4, 5, 18	finds 6711	1 |- ( x e. om -> ph )

Syntax hints:   -> wi 4   [wsb 1726   e. wcel 1804   [.wsbc 3313   (/)c0 3770   suc csuc 4870   omcom 6685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-om 6686

This theorem is referenced by: (None)