Metamath Proof Explorer


Theorem wfis2

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

Ref Expression
Hypotheses wfis2.1
|- R We A
wfis2.2
|- R Se A
wfis2.3
|- ( y = z -> ( ph <-> ps ) )
wfis2.4
|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
Assertion wfis2
|- ( y e. A -> ph )

Proof

Step Hyp Ref Expression
1 wfis2.1
 |-  R We A
2 wfis2.2
 |-  R Se A
3 wfis2.3
 |-  ( y = z -> ( ph <-> ps ) )
4 wfis2.4
 |-  ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
5 3 4 wfis2g
 |-  ( ( R We A /\ R Se A ) -> A. y e. A ph )
6 1 2 5 mp2an
 |-  A. y e. A ph
7 6 rspec
 |-  ( y e. A -> ph )