Theorem tfis2f 4650

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 1815	. . . 4 |- ( [ y / x ] ph <-> ps )
4	3	ralbii 2514	. . 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 4649	1 |- ( x e. On -> ph )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1484   [wsb 1786   e. wcel 2178   A.wral 2486   Oncon0 4428

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433

This theorem is referenced by:  tfis2  4651  tfri3  6476