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Theorem finds2 6493
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 4410 . . . . . 6  |-  (/)  e.  _V
3 finds2.1 . . . . . . 7  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
43imbi2d 316 . . . . . 6  |-  ( x  =  (/)  ->  ( ( ta  ->  ph )  <->  ( ta  ->  ps ) ) )
52, 4elab 3095 . . . . 5  |-  ( (/)  e.  { x  |  ( ta  ->  ph ) }  <-> 
( ta  ->  ps ) )
61, 5mpbir 209 . . . 4  |-  (/)  e.  {
x  |  ( ta 
->  ph ) }
7 finds2.5 . . . . . . 7  |-  ( y  e.  om  ->  ( ta  ->  ( ch  ->  th ) ) )
87a2d 26 . . . . . 6  |-  ( y  e.  om  ->  (
( ta  ->  ch )  ->  ( ta  ->  th ) ) )
9 vex 2965 . . . . . . 7  |-  y  e. 
_V
10 finds2.2 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
1110imbi2d 316 . . . . . . 7  |-  ( x  =  y  ->  (
( ta  ->  ph )  <->  ( ta  ->  ch )
) )
129, 11elab 3095 . . . . . 6  |-  ( y  e.  { x  |  ( ta  ->  ph ) } 
<->  ( ta  ->  ch ) )
139sucex 6411 . . . . . . 7  |-  suc  y  e.  _V
14 finds2.3 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
1514imbi2d 316 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ( ta  ->  ph )  <->  ( ta  ->  th ) ) )
1613, 15elab 3095 . . . . . 6  |-  ( suc  y  e.  { x  |  ( ta  ->  ph ) }  <->  ( ta  ->  th ) )
178, 12, 163imtr4g 270 . . . . 5  |-  ( y  e.  om  ->  (
y  e.  { x  |  ( ta  ->  ph ) }  ->  suc  y  e.  { x  |  ( ta  ->  ph ) } ) )
1817rgen 2771 . . . 4  |-  A. y  e.  om  ( y  e. 
{ x  |  ( ta  ->  ph ) }  ->  suc  y  e.  { x  |  ( ta 
->  ph ) } )
19 peano5 6488 . . . 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 665 . . 3  |-  om  C_  { x  |  ( ta  ->  ph ) }
2120sseli 3340 . 2  |-  ( x  e.  om  ->  x  e.  { x  |  ( ta  ->  ph ) } )
22 abid 2421 . 2  |-  ( x  e.  { x  |  ( ta  ->  ph ) } 
<->  ( ta  ->  ph )
)
2321, 22sylib 196 1  |-  ( x  e.  om  ->  ( ta  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1362    e. wcel 1755   {cab 2419   A.wral 2705    C_ wss 3316   (/)c0 3625   suc csuc 4708   omcom 6465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-tr 4374  df-eprel 4619  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-om 6466
This theorem is referenced by:  finds1  6494  onnseq  6791  nnacl  7038  nnmcl  7039  nnecl  7040  nnacom  7044  nnaass  7049  nndi  7050  nnmass  7051  nnmsucr  7052  nnmcom  7053  nnmordi  7058  omsmolem  7080  isinf  7514  unblem2  7553  fiint  7576  dffi3  7669  card2inf  7758  cantnfle  7867  cantnflt  7868  cantnflem1  7885  cantnfleOLD  7897  cantnfltOLD  7898  cantnflem1OLD  7908  cnfcom  7921  cnfcomOLD  7929  trcl  7936  fseqenlem1  8182  infpssrlem3  8462  fin23lem26  8482  axdc3lem2  8608  axdc4lem  8612  axdclem2  8677  wunr1om  8873  wuncval2  8901  tskr1om  8921  grothomex  8983  peano5nni  10312  neibastop2lem  28422
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