Theorem tfis3 4563

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 4562	. 2 |- ( x e. On -> ph )
5	1, 4	vtoclga 2792	1 |- ( A e. On -> ch )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   Oncon0 4341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346

This theorem is referenced by:  tfisi  4564  tfrlemi1  6300  tfr1onlemaccex  6316  tfrcllemaccex  6329  tfrcl  6332