Theorem frins2f

Description: Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2011) (Revised by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Hypotheses	frins2f.1	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
	frins2f.2	|- F/ y ps
	frins2f.3	|- ( y = z -> ( ph <-> ps ) )
Assertion	frins2f	|- ( ( R Fr A /\ R Se A ) -> A. y e. A ph )

Proof

Step	Hyp	Ref	Expression
1		frins2f.1	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
2		frins2f.2	|- F/ y ps
3		frins2f.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	7 1	biimtrid	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) )
9	8	frinsg	|- ( ( R Fr A /\ R Se A ) -> A. y e. A ph )