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Theorem finds2 6525
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, 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 4443 . . . . . 6  |-  (/)  e.  _V
3 finds2.1 . . . . . . 7  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
43imbi2d 316 . . . . . 6  |-  ( x  =  (/)  ->  ( ( ta  ->  ph )  <->  ( ta  ->  ps ) ) )
52, 4elab 3127 . . . . 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 2996 . . . . . . 7  |-  y  e. 
_V
10 finds2.2 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
1110imbi2d 316 . . . . . . 7  |-  ( x  =  y  ->  (
( ta  ->  ph )  <->  ( ta  ->  ch )
) )
129, 11elab 3127 . . . . . 6  |-  ( y  e.  { x  |  ( ta  ->  ph ) } 
<->  ( ta  ->  ch ) )
139sucex 6443 . . . . . . 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 3127 . . . . . 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 2802 . . . 4  |-  A. y  e.  om  ( y  e. 
{ x  |  ( ta  ->  ph ) }  ->  suc  y  e.  { x  |  ( ta 
->  ph ) } )
19 peano5 6520 . . . 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 672 . . 3  |-  om  C_  { x  |  ( ta  ->  ph ) }
2120sseli 3373 . 2  |-  ( x  e.  om  ->  x  e.  { x  |  ( ta  ->  ph ) } )
22 abid 2431 . 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 1369    e. wcel 1756   {cab 2429   A.wral 2736    C_ wss 3349   (/)c0 3658   suc csuc 4742   omcom 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-tr 4407  df-eprel 4653  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-om 6498
This theorem is referenced by:  finds1  6526  onnseq  6826  nnacl  7071  nnmcl  7072  nnecl  7073  nnacom  7077  nnaass  7082  nndi  7083  nnmass  7084  nnmsucr  7085  nnmcom  7086  nnmordi  7091  omsmolem  7113  isinf  7547  unblem2  7586  fiint  7609  dffi3  7702  card2inf  7791  cantnfle  7900  cantnflt  7901  cantnflem1  7918  cantnfleOLD  7930  cantnfltOLD  7931  cantnflem1OLD  7941  cnfcom  7954  cnfcomOLD  7962  trcl  7969  fseqenlem1  8215  infpssrlem3  8495  fin23lem26  8515  axdc3lem2  8641  axdc4lem  8645  axdclem2  8710  wunr1om  8907  wuncval2  8935  tskr1om  8955  grothomex  9017  peano5nni  10346  neibastop2lem  28607
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