Theorem tfis3 4622

Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)

Hypotheses

Ref	Expression
tfis3.1	|- ( x = y -> ( ph <-> ps ) )
tfis3.2	|- ( x = A -> ( ph <-> ch ) )
tfis3.3	|- ( x e. On -> ( A. y e. x ps -> ph ) )

Assertion

Ref	Expression
tfis3	|- ( A e. On -> ch )

Distinct variable groups:   ps, x   ph, y   ch, x   x, A   x, y

Allowed substitution hints:   ph( x)   ps( y)   ch( y)   A( y)

Proof of Theorem tfis3

Step	Hyp	Ref	Expression
1		tfis3.2	. 2 |- ( x = A -> ( ph <-> ch ) )
2		tfis3.1	. . 3 |- ( x = y -> ( ph <-> ps ) )
3		tfis3.3	. . 3 |- ( x e. On -> ( A. y e. x ps -> ph ) )
4	2, 3	tfis2 4621	. 2 |- ( x e. On -> ph )
5	1, 4	vtoclga 2830	1 |- ( A e. On -> ch )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   Oncon0 4398

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-in 3163  df-ss 3170  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403

This theorem is referenced by:  tfisi  4623  tfrlemi1  6390  tfr1onlemaccex  6406  tfrcllemaccex  6419  tfrcl  6422