Theorem findes 4651

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 3001	. 2 |- ( z = (/) -> ( [ z / x ] ph <-> [. (/) / x ]. ph ) )
2		sbequ 1863	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
3		dfsbcq2 3001	. 2 |- ( z = suc y -> ( [ z / x ] ph <-> [. suc y / x ]. ph ) )
4		sbequ12r 1795	. 2 |- ( z = x -> ( [ z / x ] ph <-> ph ) )
5		findes.1	. 2 |- [. (/) / x ]. ph
6		nfv 1551	. . . 4 |- F/ x y e. om
7		nfs1v 1967	. . . . 5 |- F/ x [ y / x ] ph
8		nfsbc1v 3017	. . . . 5 |- F/ x [. suc y / x ]. ph
9	7, 8	nfim 1595	. . . 4 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
10	6, 9	nfim 1595	. . 3 |- F/ x ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) )
11		eleq1 2268	. . . 4 |- ( x = y -> ( x e. om <-> y e. om ) )
12		sbequ12 1794	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
13		suceq 4449	. . . . . 6 |- ( x = y -> suc x = suc y )
14		dfsbcq 3000	. . . . . 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 1780	. 2 |- ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) )
20	1, 2, 3, 4, 5, 19	finds 4648	1 |- ( x e. om -> ph )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   [wsb 1785   e. wcel 2176   [.wsbc 2998   (/)c0 3460   suc csuc 4412   omcom 4638

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4418  df-iom 4639

This theorem is referenced by: (None)