Theorem finds2 7094

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 4790	. . . . . 6 |- (/) e. _V
3		finds2.1	. . . . . . 7 |- ( x = (/) -> ( ph <-> ps ) )
4	3	imbi2d 330	. . . . . 6 |- ( x = (/) -> ( ( ta -> ph ) <-> ( ta -> ps ) ) )
5	2, 4	elab 3350	. . . . 5 |- ( (/) e. { x | ( ta -> ph ) } <-> ( ta -> ps ) )
6	1, 5	mpbir 221	. . . 4 |- (/) e. { x | ( ta -> ph ) }
7		finds2.5	. . . . . . 7 |- ( y e. om -> ( ta -> ( ch -> th ) ) )
8	7	a2d 29	. . . . . 6 |- ( y e. om -> ( ( ta -> ch ) -> ( ta -> th ) ) )
9		vex 3203	. . . . . . 7 |- y e. _V
10		finds2.2	. . . . . . . 8 |- ( x = y -> ( ph <-> ch ) )
11	10	imbi2d 330	. . . . . . 7 |- ( x = y -> ( ( ta -> ph ) <-> ( ta -> ch ) ) )
12	9, 11	elab 3350	. . . . . 6 |- ( y e. { x | ( ta -> ph ) } <-> ( ta -> ch ) )
13	9	sucex 7011	. . . . . . 7 |- suc y e. _V
14		finds2.3	. . . . . . . 8 |- ( x = suc y -> ( ph <-> th ) )
15	14	imbi2d 330	. . . . . . 7 |- ( x = suc y -> ( ( ta -> ph ) <-> ( ta -> th ) ) )
16	13, 15	elab 3350	. . . . . 6 |- ( suc y e. { x | ( ta -> ph ) } <-> ( ta -> th ) )
17	8, 12, 16	3imtr4g 285	. . . . 5 |- ( y e. om -> ( y e. { x | ( ta -> ph ) } -> suc y e. { x | ( ta -> ph ) } ) )
18	17	rgen 2922	. . . 4 |- A. y e. om ( y e. { x | ( ta -> ph ) } -> suc y e. { x | ( ta -> ph ) } )
19		peano5 7089	. . . 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 708	. . 3 |- om C_ { x | ( ta -> ph ) }
21	20	sseli 3599	. 2 |- ( x e. om -> x e. { x | ( ta -> ph ) } )
22		abid 2610	. 2 |- ( x e. { x | ( ta -> ph ) } <-> ( ta -> ph ) )
23	21, 22	sylib 208	1 |- ( x e. om -> ( ta -> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   C_ wss 3574   (/)c0 3915   suc csuc 5725   omcom 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066

This theorem is referenced by:  finds1  7095  onnseq  7441  nnacl  7691  nnmcl  7692  nnecl  7693  nnacom  7697  nnaass  7702  nndi  7703  nnmass  7704  nnmsucr  7705  nnmcom  7706  nnmordi  7711  omsmolem  7733  isinf  8173  unblem2  8213  fiint  8237  dffi3  8337  card2inf  8460  cantnfle  8568  cantnflt  8569  cantnflem1  8586  cnfcom  8597  trcl  8604  fseqenlem1  8847  infpssrlem3  9127  fin23lem26  9147  axdc3lem2  9273  axdc4lem  9277  axdclem2  9342  wunr1om  9541  wuncval2  9569  tskr1om  9589  grothomex  9651  peano5nni  11023  neibastop2lem  32355  finxpreclem6  33233