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 )