Theorem finds 4343

Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.)

Hypotheses

Ref	Expression
finds.1	|- ( x = (/) -> ( ph <-> ps ) )
finds.2	|- ( x = y -> ( ph <-> ch ) )
finds.3	|- ( x = suc y -> ( ph <-> th ) )
finds.4	|- ( x = A -> ( ph <-> ta ) )
finds.5	|- ps
finds.6	|- ( y e. om -> ( ch -> th ) )

Assertion

Ref	Expression
finds	|- ( A e. om -> ta )

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

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

Proof of Theorem finds

Step	Hyp	Ref	Expression
1		finds.5	. . . . 5 |- ps
2		0ex 3907	. . . . . 6 |- (/) e. _V
3		finds.1	. . . . . 6 |- ( x = (/) -> ( ph <-> ps ) )
4	2, 3	elab 2739	. . . . 5 |- ( (/) e. { x | ph } <-> ps )
5	1, 4	mpbir 144	. . . 4 |- (/) e. { x | ph }
6		finds.6	. . . . . 6 |- ( y e. om -> ( ch -> th ) )
7		vex 2605	. . . . . . 7 |- y e. _V
8		finds.2	. . . . . . 7 |- ( x = y -> ( ph <-> ch ) )
9	7, 8	elab 2739	. . . . . 6 |- ( y e. { x | ph } <-> ch )
10	7	sucex 4245	. . . . . . 7 |- suc y e. _V
11		finds.3	. . . . . . 7 |- ( x = suc y -> ( ph <-> th ) )
12	10, 11	elab 2739	. . . . . 6 |- ( suc y e. { x | ph } <-> th )
13	6, 9, 12	3imtr4g 203	. . . . 5 |- ( y e. om -> ( y e. { x | ph } -> suc y e. { x | ph } ) )
14	13	rgen 2417	. . . 4 |- A. y e. om ( y e. { x | ph } -> suc y e. { x | ph } )
15		peano5 4341	. . . 4 |- ( ( (/) e. { x | ph } /\ A. y e. om ( y e. { x | ph } -> suc y e. { x | ph } ) ) -> om C_ { x | ph } )
16	5, 14, 15	mp2an 417	. . 3 |- om C_ { x | ph }
17	16	sseli 2996	. 2 |- ( A e. om -> A e. { x | ph } )
18		finds.4	. . 3 |- ( x = A -> ( ph <-> ta ) )
19	18	elabg 2740	. 2 |- ( A e. om -> ( A e. { x | ph } <-> ta ) )
20	17, 19	mpbid 145	1 |- ( A e. om -> ta )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1285   e. wcel 1434   {cab 2068   A.wral 2349   C_ wss 2974   (/)c0 3252   suc csuc 4122   omcom 4333

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-iinf 4331

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-uni 3604  df-int 3639  df-suc 4128  df-iom 4334

This theorem is referenced by:  findes  4346  nn0suc  4347  elnn  4348  ordom  4349  nndceq0  4359  0elnn  4360  nna0r  6115  nnm0r  6116  nnsucelsuc  6128  nneneq  6382  php5  6383  php5dom  6388  frec2uzltd  9474  frecuzrdgg  9487  iseqvalt  9521  omgadd  9815