Theorem wfis2fg

Description: Well-Ordered Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011) (Proof shortened by Scott Fenton, 17-Nov-2024)

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

Proof

Step	Hyp	Ref	Expression
1		wfis2fg.1	|- F/ y ps
2		wfis2fg.2	|- ( y = z -> ( ph <-> ps ) )
3		wfis2fg.3	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
4		wefr	|- ( R We A -> R Fr A )
5	4	adantr	|- ( ( R We A /\ R Se A ) -> R Fr A )
6		weso	|- ( R We A -> R Or A )
7		sopo	|- ( R Or A -> R Po A )
8	6 7	syl	|- ( R We A -> R Po A )
9	8	adantr	|- ( ( R We A /\ R Se A ) -> R Po A )
10		simpr	|- ( ( R We A /\ R Se A ) -> R Se A )
11	3 1 2	frpoins2fg	|- ( ( R Fr A /\ R Po A /\ R Se A ) -> A. y e. A ph )
12	5 9 10 11	syl3anc	|- ( ( R We A /\ R Se A ) -> A. y e. A ph )