Theorem frege88

Description: Commuted form of frege87 . Proposition 88 of Frege1879 p. 67. (Contributed by RP, 1-Jul-2020) (Revised by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege87.x	|- X e. U
	frege87.y	|- Y e. V
	frege87.z	|- Z e. W
	frege87.r	|- R e. S
	frege87.a	|- A e. B
Assertion	frege88	|- ( Y R Z -> ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> Z e. A ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege87.x	|- X e. U
2		frege87.y	|- Y e. V
3		frege87.z	|- Z e. W
4		frege87.r	|- R e. S
5		frege87.a	|- A e. B
6	1 2 3 4 5	frege87	|- ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) )
7		frege15	|- ( ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) ) -> ( Y R Z -> ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> Z e. A ) ) ) ) )
8	6 7	ax-mp	|- ( Y R Z -> ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> Z e. A ) ) ) )