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Theorem finds1 4486
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 2117 . 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 4485 . 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 1316    e. wcel 1465   (/)c0 3333   suc csuc 4257   omcom 4474
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-suc 4263  df-iom 4475
This theorem is referenced by:  findcard  6750  findcard2  6751  findcard2s  6752
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