Theorem finds2 4694

Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)

Hypotheses

Ref	Expression
finds2.1	|- ( x = (/) -> ( ph <-> ps ) )
finds2.2	|- ( x = y -> ( ph <-> ch ) )
finds2.3	|- ( x = suc y -> ( ph <-> th ) )
finds2.4	|- ( ta -> ps )
finds2.5	|- ( y e. om -> ( ta -> ( ch -> th ) ) )

Assertion

Ref	Expression
finds2	|- ( x e. om -> ( ta -> ph ) )

Distinct variable groups:   x, y, ta   ps, x   ch, x   th, x   ph, y

Allowed substitution hints:   ph( x)   ps( y)   ch( y)   th( y)

Proof of Theorem finds2

Step	Hyp	Ref	Expression
1		finds2.4	. . . . 5 |- ( ta -> ps )
2		0ex 4211	. . . . . 6 |- (/) e. _V
3		finds2.1	. . . . . . 7 |- ( x = (/) -> ( ph <-> ps ) )
4	3	imbi2d 230	. . . . . 6 |- ( x = (/) -> ( ( ta -> ph ) <-> ( ta -> ps ) ) )
5	2, 4	elab 2947	. . . . 5 |- ( (/) e. { x | ( ta -> ph ) } <-> ( ta -> ps ) )
6	1, 5	mpbir 146	. . . 4 |- (/) e. { x | ( ta -> ph ) }
7		finds2.5	. . . . . . 7 |- ( y e. om -> ( ta -> ( ch -> th ) ) )
8	7	a2d 26	. . . . . 6 |- ( y e. om -> ( ( ta -> ch ) -> ( ta -> th ) ) )
9		vex 2802	. . . . . . 7 |- y e. _V
10		finds2.2	. . . . . . . 8 |- ( x = y -> ( ph <-> ch ) )
11	10	imbi2d 230	. . . . . . 7 |- ( x = y -> ( ( ta -> ph ) <-> ( ta -> ch ) ) )
12	9, 11	elab 2947	. . . . . 6 |- ( y e. { x | ( ta -> ph ) } <-> ( ta -> ch ) )
13	9	sucex 4592	. . . . . . 7 |- suc y e. _V
14		finds2.3	. . . . . . . 8 |- ( x = suc y -> ( ph <-> th ) )
15	14	imbi2d 230	. . . . . . 7 |- ( x = suc y -> ( ( ta -> ph ) <-> ( ta -> th ) ) )
16	13, 15	elab 2947	. . . . . 6 |- ( suc y e. { x | ( ta -> ph ) } <-> ( ta -> th ) )
17	8, 12, 16	3imtr4g 205	. . . . 5 |- ( y e. om -> ( y e. { x | ( ta -> ph ) } -> suc y e. { x | ( ta -> ph ) } ) )
18	17	rgen 2583	. . . 4 |- A. y e. om ( y e. { x | ( ta -> ph ) } -> suc y e. { x | ( ta -> ph ) } )
19		peano5 4691	. . . 4 |- ( ( (/) e. { x | ( ta -> ph ) } /\ A. y e. om ( y e. { x | ( ta -> ph ) } -> suc y e. { x | ( ta -> ph ) } ) ) -> om C_ { x | ( ta -> ph ) } )
20	6, 18, 19	mp2an 426	. . 3 |- om C_ { x | ( ta -> ph ) }
21	20	sseli 3220	. 2 |- ( x e. om -> x e. { x | ( ta -> ph ) } )
22		abid 2217	. 2 |- ( x e. { x | ( ta -> ph ) } <-> ( ta -> ph ) )
23	21, 22	sylib 122	1 |- ( x e. om -> ( ta -> ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   C_ wss 3197   (/)c0 3491   suc csuc 4457   omcom 4683

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-iinf 4681

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-int 3924  df-suc 4463  df-iom 4684

This theorem is referenced by:  finds1  4695  frecrdg  6565  nnacl  6639  nnmcl  6640  nnacom  6643  nnaass  6644  nndi  6645  nnmass  6646  nnmsucr  6647  nnmcom  6648  nnsucsssuc  6651  nntri3or  6652  nnaordi  6667  nnaword  6670  nnmordi  6675  nnaordex  6687  fiintim  7109  prarloclem3  7700  frec2uzuzd  10641  frec2uzrdg  10648