Theorem finds2 4693

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 4591	. . . . . . 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 4690	. . . 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 4456   omcom 4682

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 4258  ax-pr 4293  ax-un 4524  ax-iinf 4680

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 4462  df-iom 4683

This theorem is referenced by:  finds1  4694  frecrdg  6554  nnacl  6626  nnmcl  6627  nnacom  6630  nnaass  6631  nndi  6632  nnmass  6633  nnmsucr  6634  nnmcom  6635  nnsucsssuc  6638  nntri3or  6639  nnaordi  6654  nnaword  6657  nnmordi  6662  nnaordex  6674  fiintim  7093  prarloclem3  7684  frec2uzuzd  10624  frec2uzrdg  10631