Theorem tfis2f 4682

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 1839	. . . 4 |- ( [ y / x ] ph <-> ps )
4	3	ralbii 2538	. . 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 4681	1 |- ( x e. On -> ph )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1508   [wsb 1810   e. wcel 2202   A.wral 2510   Oncon0 4460

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465

This theorem is referenced by:  tfis2  4683  tfri3  6532