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Theorem finds1 4576
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 2253 . 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 4575 . 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 3362   suc csuc 4287   omcom 4547
This theorem is referenced by:  findcard  6982  findcard2  6983  pwfi  7035  alephfplem3  7617  pwsdompw  7714  hsmexlem4  7939  expus  24531
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548
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