Theorem frege85

Description: Commuted form of frege77 . Proposition 85 of Frege1879 p. 66. (Contributed by RP, 1-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege84.x	|- X e. U
	frege84.y	|- Y e. V
	frege84.r	|- R e. W
	frege84.a	|- A e. B
Assertion	frege85	|- ( X ( t+ ` R ) Y -> ( A. z ( X R z -> z e. A ) -> ( R hereditary A -> Y e. A ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege84.x	|- X e. U
2		frege84.y	|- Y e. V
3		frege84.r	|- R e. W
4		frege84.a	|- A e. B
5	1 2 3 4	frege77	|- ( X ( t+ ` R ) Y -> ( R hereditary A -> ( A. z ( X R z -> z e. A ) -> Y e. A ) ) )
6		frege12	|- ( ( X ( t+ ` R ) Y -> ( R hereditary A -> ( A. z ( X R z -> z e. A ) -> Y e. A ) ) ) -> ( X ( t+ ` R ) Y -> ( A. z ( X R z -> z e. A ) -> ( R hereditary A -> Y e. A ) ) ) )
7	5 6	ax-mp	|- ( X ( t+ ` R ) Y -> ( A. z ( X R z -> z e. A ) -> ( R hereditary A -> Y e. A ) ) )