Theorem findes 4600

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	|- [. (/) / 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 2965	. 2 |- ( z = (/) -> ( [ z / x ] ph <-> [. (/) / x ]. ph ) )
2		sbequ 1840	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
3		dfsbcq2 2965	. 2 |- ( z = suc y -> ( [ z / x ] ph <-> [. suc y / x ]. ph ) )
4		sbequ12r 1772	. 2 |- ( z = x -> ( [ z / x ] ph <-> ph ) )
5		findes.1	. 2 |- [. (/) / x ]. ph
6		nfv 1528	. . . 4 |- F/ x y e. om
7		nfs1v 1939	. . . . 5 |- F/ x [ y / x ] ph
8		nfsbc1v 2981	. . . . 5 |- F/ x [. suc y / x ]. ph
9	7, 8	nfim 1572	. . . 4 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
10	6, 9	nfim 1572	. . 3 |- F/ x ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) )
11		eleq1 2240	. . . 4 |- ( x = y -> ( x e. om <-> y e. om ) )
12		sbequ12 1771	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
13		suceq 4400	. . . . . 6 |- ( x = y -> suc x = suc y )
14		dfsbcq 2964	. . . . . 6 |- ( suc x = suc y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) )
15	13, 14	syl 14	. . . . 5 |- ( x = y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) )
16	12, 15	imbi12d 234	. . . 4 |- ( x = y -> ( ( ph -> [. suc x / x ]. ph ) <-> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) )
17	11, 16	imbi12d 234	. . 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 1757	. 2 |- ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) )
20	1, 2, 3, 4, 5, 19	finds 4597	1 |- ( x e. om -> ph )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   [wsb 1762   e. wcel 2148   [.wsbc 2962   (/)c0 3422   suc csuc 4363   omcom 4587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-iinf 4585

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3809  df-int 3844  df-suc 4369  df-iom 4588

This theorem is referenced by: (None)