Theorem tfis3 4708

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 4707	. 2 |- ( x e. On -> ph )
5	1, 4	vtoclga 2881	1 |- ( A e. On -> ch )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203   A.wral 2520   Oncon0 4484

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-in 3217  df-ss 3224  df-uni 3915  df-tr 4209  df-iord 4487  df-on 4489

This theorem is referenced by:  tfisi  4709  tfrlemi1  6563  tfr1onlemaccex  6579  tfrcllemaccex  6592  tfrcl  6595