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Theorem finds2 4686
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 hypothesis. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)
Hypotheses
Ref Expression
finds2.1  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
finds2.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
finds2.3  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
finds2.4  |-  ( ta 
->  ps )
finds2.5  |-  ( y  e.  om  ->  ( ta  ->  ( ch  ->  th ) ) )
Assertion
Ref Expression
finds2  |-  ( x  e.  om  ->  ( ta  ->  ph ) )
Distinct variable groups:    x, y, ta    ps, x    ch, x    th, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)    ch( y)    th( y)

Proof of Theorem finds2
StepHypRef Expression
1 finds2.4 . . . . 5  |-  ( ta 
->  ps )
2 0ex 4152 . . . . . 6  |-  (/)  e.  _V
3 finds2.1 . . . . . . 7  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
43imbi2d 307 . . . . . 6  |-  ( x  =  (/)  ->  ( ( ta  ->  ph )  <->  ( ta  ->  ps ) ) )
52, 4elab 2916 . . . . 5  |-  ( (/)  e.  { x  |  ( ta  ->  ph ) }  <-> 
( ta  ->  ps ) )
61, 5mpbir 200 . . . 4  |-  (/)  e.  {
x  |  ( ta 
->  ph ) }
7 finds2.5 . . . . . . 7  |-  ( y  e.  om  ->  ( ta  ->  ( ch  ->  th ) ) )
87a2d 23 . . . . . 6  |-  ( y  e.  om  ->  (
( ta  ->  ch )  ->  ( ta  ->  th ) ) )
9 vex 2793 . . . . . . 7  |-  y  e. 
_V
10 finds2.2 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
1110imbi2d 307 . . . . . . 7  |-  ( x  =  y  ->  (
( ta  ->  ph )  <->  ( ta  ->  ch )
) )
129, 11elab 2916 . . . . . 6  |-  ( y  e.  { x  |  ( ta  ->  ph ) } 
<->  ( ta  ->  ch ) )
139sucex 4604 . . . . . . 7  |-  suc  y  e.  _V
14 finds2.3 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
1514imbi2d 307 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ( ta  ->  ph )  <->  ( ta  ->  th ) ) )
1613, 15elab 2916 . . . . . 6  |-  ( suc  y  e.  { x  |  ( ta  ->  ph ) }  <->  ( ta  ->  th ) )
178, 12, 163imtr4g 261 . . . . 5  |-  ( y  e.  om  ->  (
y  e.  { x  |  ( ta  ->  ph ) }  ->  suc  y  e.  { x  |  ( ta  ->  ph ) } ) )
1817rgen 2610 . . . 4  |-  A. y  e.  om  ( y  e. 
{ x  |  ( ta  ->  ph ) }  ->  suc  y  e.  { x  |  ( ta 
->  ph ) } )
19 peano5 4681 . . . 4  |-  ( (
(/)  e.  { x  |  ( ta  ->  ph ) }  /\  A. y  e.  om  (
y  e.  { x  |  ( ta  ->  ph ) }  ->  suc  y  e.  { x  |  ( ta  ->  ph ) } ) )  ->  om  C_  { x  |  ( ta  ->  ph ) } )
206, 18, 19mp2an 653 . . 3  |-  om  C_  { x  |  ( ta  ->  ph ) }
2120sseli 3178 . 2  |-  ( x  e.  om  ->  x  e.  { x  |  ( ta  ->  ph ) } )
22 abid 2273 . 2  |-  ( x  e.  { x  |  ( ta  ->  ph ) } 
<->  ( ta  ->  ph )
)
2321, 22sylib 188 1  |-  ( x  e.  om  ->  ( ta  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1625    e. wcel 1686   {cab 2271   A.wral 2545    C_ wss 3154   (/)c0 3457   suc csuc 4396   omcom 4658
This theorem is referenced by:  finds1  4687  onnseq  6363  nnacl  6611  nnmcl  6612  nnecl  6613  nnacom  6617  nnaass  6622  nndi  6623  nnmass  6624  nnmsucr  6625  nnmcom  6626  nnmordi  6631  omsmolem  6653  isinf  7078  unblem2  7112  fiint  7135  dffi3  7186  card2inf  7271  cantnfle  7374  cantnflt  7375  cantnflem1  7393  cnfcom  7405  trcl  7412  fseqenlem1  7653  infpssrlem3  7933  fin23lem26  7953  axdc3lem2  8079  axdc4lem  8083  axdclem2  8149  wunr1om  8343  wuncval2  8371  tskr1om  8391  grothomex  8453  peano5nni  9751  neibastop2lem  26320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-tr 4116  df-eprel 4307  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659
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