Theorem finds2 4523

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 4063	. . . . . 6 |- (/) e. _V
3		finds2.1	. . . . . . 7 |- ( x = (/) -> ( ph <-> ps ) )
4	3	imbi2d 229	. . . . . 6 |- ( x = (/) -> ( ( ta -> ph ) <-> ( ta -> ps ) ) )
5	2, 4	elab 2832	. . . . 5 |- ( (/) e. { x | ( ta -> ph ) } <-> ( ta -> ps ) )
6	1, 5	mpbir 145	. . . 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 2692	. . . . . . 7 |- y e. _V
10		finds2.2	. . . . . . . 8 |- ( x = y -> ( ph <-> ch ) )
11	10	imbi2d 229	. . . . . . 7 |- ( x = y -> ( ( ta -> ph ) <-> ( ta -> ch ) ) )
12	9, 11	elab 2832	. . . . . 6 |- ( y e. { x | ( ta -> ph ) } <-> ( ta -> ch ) )
13	9	sucex 4423	. . . . . . 7 |- suc y e. _V
14		finds2.3	. . . . . . . 8 |- ( x = suc y -> ( ph <-> th ) )
15	14	imbi2d 229	. . . . . . 7 |- ( x = suc y -> ( ( ta -> ph ) <-> ( ta -> th ) ) )
16	13, 15	elab 2832	. . . . . 6 |- ( suc y e. { x | ( ta -> ph ) } <-> ( ta -> th ) )
17	8, 12, 16	3imtr4g 204	. . . . 5 |- ( y e. om -> ( y e. { x | ( ta -> ph ) } -> suc y e. { x | ( ta -> ph ) } ) )
18	17	rgen 2488	. . . 4 |- A. y e. om ( y e. { x | ( ta -> ph ) } -> suc y e. { x | ( ta -> ph ) } )
19		peano5 4520	. . . 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 423	. . 3 |- om C_ { x | ( ta -> ph ) }
21	20	sseli 3098	. 2 |- ( x e. om -> x e. { x | ( ta -> ph ) } )
22		abid 2128	. 2 |- ( x e. { x | ( ta -> ph ) } <-> ( ta -> ph ) )
23	21, 22	sylib 121	1 |- ( x e. om -> ( ta -> ph ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   e. wcel 1481   {cab 2126   A.wral 2417   C_ wss 3076   (/)c0 3368   suc csuc 4295   omcom 4512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-int 3780  df-suc 4301  df-iom 4513

This theorem is referenced by:  finds1  4524  frecrdg  6313  nnacl  6384  nnmcl  6385  nnacom  6388  nnaass  6389  nndi  6390  nnmass  6391  nnmsucr  6392  nnmcom  6393  nnsucsssuc  6396  nntri3or  6397  nnaordi  6412  nnaword  6415  nnmordi  6420  nnaordex  6431  fiintim  6825  prarloclem3  7329  frec2uzuzd  10206  frec2uzrdg  10213