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Theorem tfis3 6691
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
Hypotheses
Ref Expression
tfis3.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
tfis3.2  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
tfis3.3  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
Assertion
Ref Expression
tfis3  |-  ( A  e.  On  ->  ch )
Distinct variable groups:    ps, x    ph, y    ch, x    x, A    x, y
Allowed substitution hints:    ph( x)    ps( y)    ch( y)    A( y)

Proof of Theorem tfis3
StepHypRef Expression
1 tfis3.2 . 2  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
2 tfis3.1 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3 tfis3.3 . . 3  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
42, 3tfis2 6690 . 2  |-  ( x  e.  On  ->  ph )
51, 4vtoclga 3173 1  |-  ( A  e.  On  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    e. wcel 1819   A.wral 2807   Oncon0 4887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891
This theorem is referenced by:  tfisi  6692  tfinds  6693  tfrlem1  7063  ordtypelem7  7967  rankonidlem  8263  tcrank  8319  infxpenlem  8408  alephle  8486  dfac12lem3  8542  ttukeylem5  8910  ttukeylem6  8911  tskord  9175  grudomon  9212  aomclem6  31209
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