Theorem frins

Description: Founded Induction Schema. If a property passes from all elements less than y of a founded class A to y itself (induction hypothesis), then the property holds for all elements of A . (Contributed by Scott Fenton, 6-Feb-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	frins.1	|- R Fr A
	frins.2	|- R Se A
	frins.3	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) )
Assertion	frins	|- ( y e. A -> ph )

Proof

Step	Hyp	Ref	Expression
1		frins.1	|- R Fr A
2		frins.2	|- R Se A
3		frins.3	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) )
4	3	frinsg	|- ( ( R Fr A /\ R Se A ) -> A. y e. A ph )
5	1 2 4	mp2an	|- A. y e. A ph
6	5	rspec	|- ( y e. A -> ph )