Theorem frpoins3g

Description: Founded Partial Induction schema, using implicit substitution. (Contributed by Scott Fenton, 19-Aug-2024)

	Ref	Expression
Hypotheses	frpoins3g.1	|- ( x e. A -> ( A. y e. Pred ( R , A , x ) ps -> ph ) )
	frpoins3g.2	|- ( x = y -> ( ph <-> ps ) )
	frpoins3g.3	|- ( x = B -> ( ph <-> ch ) )
Assertion	frpoins3g	|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ B e. A ) -> ch )

Proof

Step	Hyp	Ref	Expression
1		frpoins3g.1	|- ( x e. A -> ( A. y e. Pred ( R , A , x ) ps -> ph ) )
2		frpoins3g.2	|- ( x = y -> ( ph <-> ps ) )
3		frpoins3g.3	|- ( x = B -> ( ph <-> ch ) )
4	1 2	frpoins2g	|- ( ( R Fr A /\ R Po A /\ R Se A ) -> A. x e. A ph )
5	3	rspccva	|- ( ( A. x e. A ph /\ B e. A ) -> ch )
6	4 5	sylan	|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ B e. A ) -> ch )