Metamath Proof Explorer


Theorem wfis2g

Description: Well-Ordered Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011)

Ref Expression
Hypotheses wfis2g.1
|- ( y = z -> ( ph <-> ps ) )
wfis2g.2
|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
Assertion wfis2g
|- ( ( R We A /\ R Se A ) -> A. y e. A ph )

Proof

Step Hyp Ref Expression
1 wfis2g.1
 |-  ( y = z -> ( ph <-> ps ) )
2 wfis2g.2
 |-  ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
3 nfv
 |-  F/ y ps
4 3 1 2 wfis2fg
 |-  ( ( R We A /\ R Se A ) -> A. y e. A ph )