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Theorem finds1 4523
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, 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 2140 . 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 9 . . 3  |-  ( (/)  =  (/)  ->  ps )
7 finds1.5 . . . 4  |-  ( y  e.  om  ->  ( ch  ->  th ) )
87a1d 22 . . 3  |-  ( y  e.  om  ->  ( (/)  =  (/)  ->  ( ch 
->  th ) ) )
92, 3, 4, 6, 8finds2 4522 . 2  |-  ( x  e.  om  ->  ( (/)  =  (/)  ->  ph )
)
101, 9mpi 15 1  |-  ( x  e.  om  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1332    e. wcel 1481   (/)c0 3367   suc csuc 4294   omcom 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-uni 3744  df-int 3779  df-suc 4300  df-iom 4512
This theorem is referenced by:  findcard  6789  findcard2  6790  findcard2s  6791
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