Theorem finds1 4601

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 2177	. 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 4600	. 2 |- ( x e. om -> ( (/) = (/) -> ph ) )
10	1, 9	mpi 15	1 |- ( x e. om -> ph )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   (/)c0 3422   suc csuc 4365   omcom 4589

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-suc 4371  df-iom 4590

This theorem is referenced by:  findcard  6887  findcard2  6888  findcard2s  6889