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Theorem finds1 4643
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 2256 . 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 4642 . 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 1619    e. wcel 1621   (/)c0 3416   suc csuc 4352   omcom 4614
This theorem is referenced by:  findcard  7051  findcard2  7052  pwfi  7105  alephfplem3  7687  pwsdompw  7784  hsmexlem4  8009  expus  24718
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-tr 4074  df-eprel 4263  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615
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