Theorem findes 4669

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 3008	. 2 |- ( z = (/) -> ( [ z / x ] ph <-> [. (/) / x ]. ph ) )
2		sbequ 1864	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
3		dfsbcq2 3008	. 2 |- ( z = suc y -> ( [ z / x ] ph <-> [. suc y / x ]. ph ) )
4		sbequ12r 1796	. 2 |- ( z = x -> ( [ z / x ] ph <-> ph ) )
5		findes.1	. 2 |- [. (/) / x ]. ph
6		nfv 1552	. . . 4 |- F/ x y e. om
7		nfs1v 1968	. . . . 5 |- F/ x [ y / x ] ph
8		nfsbc1v 3024	. . . . 5 |- F/ x [. suc y / x ]. ph
9	7, 8	nfim 1596	. . . 4 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
10	6, 9	nfim 1596	. . 3 |- F/ x ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) )
11		eleq1 2270	. . . 4 |- ( x = y -> ( x e. om <-> y e. om ) )
12		sbequ12 1795	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
13		suceq 4467	. . . . . 6 |- ( x = y -> suc x = suc y )
14		dfsbcq 3007	. . . . . 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 1781	. 2 |- ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) )
20	1, 2, 3, 4, 5, 19	finds 4666	1 |- ( x e. om -> ph )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   [wsb 1786   e. wcel 2178   [.wsbc 3005   (/)c0 3468   suc csuc 4430   omcom 4656

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-int 3900  df-suc 4436  df-iom 4657

This theorem is referenced by: (None)