Theorem frins2f

Description: Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011) (Revised by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Hypotheses	frins2f.1	|- R Fr A
	frins2f.2	|- R Se A
	frins2f.3	|- F/ y ps
	frins2f.4	|- ( y = z -> ( ph <-> ps ) )
	frins2f.5	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
Assertion	frins2f	|- ( y e. A -> ph )

Proof

Step	Hyp	Ref	Expression
1		frins2f.1	|- R Fr A
2		frins2f.2	|- R Se A
3		frins2f.3	|- F/ y ps
4		frins2f.4	|- ( y = z -> ( ph <-> ps ) )
5		frins2f.5	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
6	5 3 4	frins2fg	|- ( ( R Fr 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 )