Metamath Proof Explorer


Theorem wfis2f

Description: Well-Ordered Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011)

Ref Expression
Hypotheses wfis2f.1
|- R We A
wfis2f.2
|- R Se A
wfis2f.3
|- F/ y ps
wfis2f.4
|- ( y = z -> ( ph <-> ps ) )
wfis2f.5
|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
Assertion wfis2f
|- ( y e. A -> ph )

Proof

Step Hyp Ref Expression
1 wfis2f.1
 |-  R We A
2 wfis2f.2
 |-  R Se A
3 wfis2f.3
 |-  F/ y ps
4 wfis2f.4
 |-  ( y = z -> ( ph <-> ps ) )
5 wfis2f.5
 |-  ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
6 3 4 5 wfis2fg
 |-  ( ( R We A /\ R Se A ) -> A. y e. A ph )
7 1 2 6 mp2an
 |-  A. y e. A ph
8 7 rspec
 |-  ( y e. A -> ph )