Theorem tfis2d 42427

Description: Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.)

Hypotheses

Ref	Expression
tfis2d.1	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )
tfis2d.2	|- ( ph -> ( x e. On -> ( A. y e. x ch -> ps ) ) )

Assertion

Ref	Expression
tfis2d	|- ( ph -> ( x e. On -> ps ) )

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

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

Proof of Theorem tfis2d

Step	Hyp	Ref	Expression
1		tfis2d.1	. . . . 5 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
2	1	com12 32	. . . 4 |- ( x = y -> ( ph -> ( ps <-> ch ) ) )
3	2	pm5.74d 262	. . 3 |- ( x = y -> ( ( ph -> ps ) <-> ( ph -> ch ) ) )
4		r19.21v 2960	. . . 4 |- ( A. y e. x ( ph -> ch ) <-> ( ph -> A. y e. x ch ) )
5		tfis2d.2	. . . . . 6 |- ( ph -> ( x e. On -> ( A. y e. x ch -> ps ) ) )
6	5	com12 32	. . . . 5 |- ( x e. On -> ( ph -> ( A. y e. x ch -> ps ) ) )
7	6	a2d 29	. . . 4 |- ( x e. On -> ( ( ph -> A. y e. x ch ) -> ( ph -> ps ) ) )
8	4, 7	syl5bi 232	. . 3 |- ( x e. On -> ( A. y e. x ( ph -> ch ) -> ( ph -> ps ) ) )
9	3, 8	tfis2 7056	. 2 |- ( x e. On -> ( ph -> ps ) )
10	9	com12 32	1 |- ( ph -> ( x e. On -> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   A.wral 2912   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by: (None)