Theorem finds1 4724

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 2232	. 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 4723	. 2 |- ( x e. om -> ( (/) = (/) -> ph ) )
10	1, 9	mpi 15	1 |- ( x e. om -> ph )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203   (/)c0 3508   suc csuc 4486   omcom 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-int 3950  df-suc 4492  df-iom 4713

This theorem is referenced by:  findcard  7145  findcard2  7146  findcard2s  7147