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Theorem tfis2f 4493
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 1764 . . . 4  |-  ( [ y  /  x ] ph 
<->  ps )
43ralbii 2439 . . 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 151 . 2  |-  ( x  e.  On  ->  ( A. y  e.  x  [ y  /  x ] ph  ->  ph ) )
76tfis 4492 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   F/wnf 1436    e. wcel 1480   [wsb 1735   A.wral 2414   Oncon0 4280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-setind 4447
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285
This theorem is referenced by:  tfis2  4494  tfri3  6257
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