Theorem finds1 4454

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, 22-Mar-2006.)

Hypotheses

Ref	Expression
finds1.1	|- ( x = (/) -> ( ph <-> ps ) )
finds1.2	|- ( x = y -> ( ph <-> ch ) )
finds1.3	|- ( x = suc y -> ( ph <-> th ) )
finds1.4	|- ps
finds1.5	|- ( y e. om -> ( ch -> th ) )

Assertion

Ref	Expression
finds1	|- ( x e. om -> ph )

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

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

Proof of Theorem finds1

Step	Hyp	Ref	Expression
1		eqid 2100	. 2 |- (/) = (/)
2		finds1.1	. . 3 |- ( x = (/) -> ( ph <-> ps ) )
3		finds1.2	. . 3 |- ( x = y -> ( ph <-> ch ) )
4		finds1.3	. . 3 |- ( x = suc y -> ( ph <-> th ) )
5		finds1.4	. . . 4 |- ps
6	5	a1i 9	. . 3 |- ( (/) = (/) -> ps )
7		finds1.5	. . . 4 |- ( y e. om -> ( ch -> th ) )
8	7	a1d 22	. . 3 |- ( y e. om -> ( (/) = (/) -> ( ch -> th ) ) )
9	2, 3, 4, 6, 8	finds2 4453	. 2 |- ( x e. om -> ( (/) = (/) -> ph ) )
10	1, 9	mpi 15	1 |- ( x e. om -> ph )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1299   e. wcel 1448   (/)c0 3310   suc csuc 4225   omcom 4442

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-int 3719  df-suc 4231  df-iom 4443

This theorem is referenced by:  findcard  6711  findcard2  6712  findcard2s  6713