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Theorem tfis2 4428
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
Hypotheses
Ref Expression
tfis2.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
tfis2.2  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
Assertion
Ref Expression
tfis2  |-  ( x  e.  On  ->  ph )
Distinct variable groups:    ps, x    ph, y    x, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem tfis2
StepHypRef Expression
1 nfv 1473 . 2  |-  F/ x ps
2 tfis2.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3 tfis2.2 . 2  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
41, 2, 3tfis2f 4427 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1445   A.wral 2370   Oncon0 4214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-setind 4381
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-in 3019  df-ss 3026  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219
This theorem is referenced by:  tfis3  4429  tfrlem1  6111  ordiso2  6808
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