Theorem finds2 4750

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 4262	. . . . . 6 |- (/) e. _V
3		finds2.1	. . . . . . 7 |- ( x = (/) -> ( ph <-> ps ) )
4	3	imbi2d 230	. . . . . 6 |- ( x = (/) -> ( ( ta -> ph ) <-> ( ta -> ps ) ) )
5	2, 4	elab 2970	. . . . 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 2824	. . . . . . 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 2970	. . . . . 6 |- ( y e. { x | ( ta -> ph ) } <-> ( ta -> ch ) )
13	9	sucex 4648	. . . . . . 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 2970	. . . . . 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 2603	. . . 4 |- A. y e. om ( y e. { x | ( ta -> ph ) } -> suc y e. { x | ( ta -> ph ) } )
19		peano5 4747	. . . 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 430	. . 3 |- om C_ { x | ( ta -> ph ) }
21	20	sseli 3244	. 2 |- ( x e. om -> x e. { x | ( ta -> ph ) } )
22		abid 2226	. 2 |- ( x e. { x | ( ta -> ph ) } <-> ( ta -> ph ) )
23	21, 22	sylib 122	1 |- ( x e. om -> ( ta -> ph ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   C_ wss 3220   (/)c0 3520   suc csuc 4512   omcom 4739

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-uni 3936  df-int 3971  df-suc 4518  df-iom 4740

This theorem is used by:  finds1  4751  frecrdg  6683  nnacl  6757  nnmcl  6758  nnacom  6761  nnaass  6762  nndi  6763  nnmass  6764  nnmsucr  6765  nnmcom  6766  nnsucsssuc  6769  nntri3or  6770  nnaordi  6785  nnaword  6788  nnmordi  6793  nnaordex  6805  fiintim  7242  prarloclem3  7870  frec2uzuzd  10860  frec2uzrdg  10867