Theorem frpoins2fg

Description: Founded Partial Induction schema, using implicit substitution. (Contributed by Scott Fenton, 24-Aug-2022)

	Ref	Expression
Hypotheses	frpoins2fg.1	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
	frpoins2fg.2	|- F/ y ps
	frpoins2fg.3	|- ( y = z -> ( ph <-> ps ) )
Assertion	frpoins2fg	|- ( ( R Fr A /\ R Po A /\ R Se A ) -> A. y e. A ph )

Proof

Step	Hyp	Ref	Expression
1		frpoins2fg.1	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
2		frpoins2fg.2	|- F/ y ps
3		frpoins2fg.3	|- ( y = z -> ( ph <-> ps ) )
4		sbsbc	|- ( [ z / y ] ph <-> [. z / y ]. ph )
5	2 3	sbiev	|- ( [ z / y ] ph <-> ps )
6	4 5	bitr3i	|- ( [. z / y ]. ph <-> ps )
7	6	ralbii	|- ( A. z e. Pred ( R , A , y ) [. z / y ]. ph <-> A. z e. Pred ( R , A , y ) ps )
8	1	adantl	|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ y e. A ) -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
9	7 8	syl5bi	|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ y e. A ) -> ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) )
10	9	frpoinsg	|- ( ( R Fr A /\ R Po A /\ R Se A ) -> A. y e. A ph )