Theorem tfis2f 4561

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 1779	. . . 4 |- ( [ y / x ] ph <-> ps )
4	3	ralbii 2472	. . 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	syl5bi 151	. 2 |- ( x e. On -> ( A. y e. x [ y / x ] ph -> ph ) )
7	6	tfis 4560	1 |- ( x e. On -> ph )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1448   [wsb 1750   e. wcel 2136   A.wral 2444   Oncon0 4341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346

This theorem is referenced by:  tfis2  4562  tfri3  6335