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Theorem tfis3 6570
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 6569 . 2  |-  ( x  e.  On  ->  ph )
51, 4vtoclga 3134 1  |-  ( A  e.  On  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   A.wral 2795   Oncon0 4819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-tr 4486  df-eprel 4732  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823
This theorem is referenced by:  tfisi  6571  tfinds  6572  ordtypelem7  7841  rankonidlem  8138  tcrank  8194  infxpenlem  8283  alephle  8361  dfac12lem3  8417  ttukeylem5  8785  ttukeylem6  8786  tskord  9050  grudomon  9087  aomclem6  29552
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