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Theorem tfis2f 4662
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 1991 . . . 4  |-  ( [ y  /  x ] ph 
<->  ps )
43ralbii 2580 . . 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 4661 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1534   [wsb 1638    e. wcel 1696   A.wral 2556   Oncon0 4408
This theorem is referenced by:  tfis2  4663  tfr3  6431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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