Theorem frege95

Description: Looking one past a pair related by transitive closure of a relation. Proposition 95 of Frege1879 p. 70. (Contributed by RP, 2-Jul-2020) (Revised by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege95.x	|- X e. U
	frege95.y	|- Y e. V
	frege95.z	|- Z e. W
	frege95.r	|- R e. A
Assertion	frege95	|- ( Y R Z -> ( X ( t+ ` R ) Y -> X ( t+ ` R ) Z ) )

Proof

Step	Hyp	Ref	Expression
1		frege95.x	|- X e. U
2		frege95.y	|- Y e. V
3		frege95.z	|- Z e. W
4		frege95.r	|- R e. A
5		vex	|- f e. _V
6	1 2 3 4 5	frege88	|- ( Y R Z -> ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. f ) -> ( R hereditary f -> Z e. f ) ) ) )
7	6	alrimdv	|- ( Y R Z -> ( X ( t+ ` R ) Y -> A. f ( A. w ( X R w -> w e. f ) -> ( R hereditary f -> Z e. f ) ) ) )
8	1 3 4	frege94	|- ( ( Y R Z -> ( X ( t+ ` R ) Y -> A. f ( A. w ( X R w -> w e. f ) -> ( R hereditary f -> Z e. f ) ) ) ) -> ( Y R Z -> ( X ( t+ ` R ) Y -> X ( t+ ` R ) Z ) ) )
9	7 8	ax-mp	|- ( Y R Z -> ( X ( t+ ` R ) Y -> X ( t+ ` R ) Z ) )