Theorem tfis2f 4675

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

Step	Hyp	Ref	Expression
1		tfis2f.1	. . . . 5 |- F/ x ps
2		tfis2f.2	. . . . 5 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	sbie 1837	. . . 4 |- ( [ y / x ] ph <-> ps )
4	3	ralbii 2536	. . 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 ) )
6	4, 5	biimtrid 152	. 2 |- ( x e. On -> ( A. y e. x [ y / x ] ph -> ph ) )
7	6	tfis 4674	1 |- ( x e. On -> ph )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1506   [wsb 1808   e. wcel 2200   A.wral 2508   Oncon0 4453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-in 3203  df-ss 3210  df-uni 3888  df-tr 4182  df-iord 4456  df-on 4458

This theorem is referenced by:  tfis2  4676  tfri3  6511