Theorem tfis2 4308

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 1421	. 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 4307	1 |- ( x e. On -> ph )

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1393   A.wral 2306   Oncon0 4100

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105

This theorem is referenced by:  tfis3  4309  tfrlem1  5923  ordiso2  6357