Metamath Proof Explorer


Theorem tfis2d

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

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

Proof

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