Theorem tfis2 4326

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

Step	Hyp	Ref	Expression
1		nfv 1461	. 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 ) )
4	1, 2, 3	tfis2f 4325	1 |- ( x e. On -> ph )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1433   A.wral 2348   Oncon0 4118

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123

This theorem is referenced by:  tfis3  4327  tfrlem1  5946  ordiso2  6446