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Theorem tfis3 6457
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 6456 . 2  |-  ( x  e.  On  ->  ph )
51, 4vtoclga 3025 1  |-  ( A  e.  On  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1362    e. wcel 1755   A.wral 2705   Oncon0 4706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-tr 4374  df-eprel 4619  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710
This theorem is referenced by:  tfisi  6458  tfinds  6459  ordtypelem7  7726  rankonidlem  8023  tcrank  8079  infxpenlem  8168  alephle  8246  dfac12lem3  8302  ttukeylem5  8670  ttukeylem6  8671  tskord  8934  grudomon  8971  aomclem6  29254
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