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Theorem finds1 4685
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, 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
StepHypRef Expression
1 eqid 2285 . 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
65a1i 12 . . 3  |-  ( (/)  =  (/)  ->  ps )
7 finds1.5 . . . 4  |-  ( y  e.  om  ->  ( ch  ->  th ) )
87a1d 24 . . 3  |-  ( y  e.  om  ->  ( (/)  =  (/)  ->  ( ch 
->  th ) ) )
92, 3, 4, 6, 8finds2 4684 . 2  |-  ( x  e.  om  ->  ( (/)  =  (/)  ->  ph )
)
101, 9mpi 18 1  |-  ( x  e.  om  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1624    e. wcel 1685   (/)c0 3457   suc csuc 4394   omcom 4656
This theorem is referenced by:  findcard  7093  findcard2  7094  pwfi  7147  alephfplem3  7729  pwsdompw  7826  hsmexlem4  8051  expus  24765
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  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 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657
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