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Theorem tfis2f 4335
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 1690 . . . 4  |-  ( [ y  /  x ] ph 
<->  ps )
43ralbii 2347 . . 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 145 . 2  |-  ( x  e.  On  ->  ( A. y  e.  x  [ y  /  x ] ph  ->  ph ) )
76tfis 4334 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   F/wnf 1365    e. wcel 1409   [wsb 1661   A.wral 2323   Oncon0 4128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133
This theorem is referenced by:  tfis2  4336  tfri3  5984
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