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Theorem tfis2f 4646
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
Hypotheses
Ref Expression
tfis2f.1  |-  F/ x ps
tfis2f.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
tfis2f.3  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
Assertion
Ref Expression
tfis2f  |-  ( x  e.  On  ->  ph )
Distinct variable groups:    ph, y    x, y
Allowed substitution hints:    ph( x)    ps( x, y)

Proof of Theorem tfis2f
StepHypRef Expression
1 tfis2f.1 . . . . 5  |-  F/ x ps
2 tfis2f.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
31, 2sbie 1978 . . . 4  |-  ( [ y  /  x ] ph 
<->  ps )
43ralbii 2567 . . 3  |-  ( A. y  e.  x  [
y  /  x ] ph 
<-> 
A. y  e.  x  ps )
5 tfis2f.3 . . 3  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
64, 5syl5bi 208 . 2  |-  ( x  e.  On  ->  ( A. y  e.  x  [ y  /  x ] ph  ->  ph ) )
76tfis 4645 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1531   [wsb 1629    e. wcel 1684   A.wral 2543   Oncon0 4392
This theorem is referenced by:  tfis2  4647  tfr3  6415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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