Theorem tfis3 4634

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 4633	. 2 |- ( x e. On -> ph )
5	1, 4	vtoclga 2839	1 |- ( A e. On -> ch )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2176   A.wral 2484   Oncon0 4410

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415

This theorem is referenced by:  tfisi  4635  tfrlemi1  6418  tfr1onlemaccex  6434  tfrcllemaccex  6447  tfrcl  6450