Theorem finds1 4343

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 2081	. 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 4342	. 2 |- ( x e. om -> ( (/) = (/) -> ph ) )
10	1, 9	mpi 15	1 |- ( x e. om -> ph )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   (/)c0 3251   suc csuc 4120   omcom 4331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-suc 4126  df-iom 4332

This theorem is referenced by:  findcard  6372  findcard2  6373  findcard2s  6374