Theorem frins3

Description: Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	frins3.1	|- ( y = z -> ( ph <-> ps ) )
	frins3.2	|- ( y = B -> ( ph <-> ch ) )
	frins3.3	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
Assertion	frins3	|- ( ( ( R Fr A /\ R Se A ) /\ B e. A ) -> ch )

Proof

Step	Hyp	Ref	Expression
1		frins3.1	|- ( y = z -> ( ph <-> ps ) )
2		frins3.2	|- ( y = B -> ( ph <-> ch ) )
3		frins3.3	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
4	3 1	frins2	|- ( ( R Fr A /\ R Se A ) -> A. y e. A ph )
5	2	rspcv	|- ( B e. A -> ( A. y e. A ph -> ch ) )
6	4 5	mpan9	|- ( ( ( R Fr A /\ R Se A ) /\ B e. A ) -> ch )