Theorem frege78

Description: Commuted form of frege77 . Proposition 78 of Frege1879 p. 63. (Contributed by RP, 1-Jul-2020) (Revised by RP, 2-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege78.x	|- X e. U
	frege78.y	|- Y e. V
	frege78.r	|- R e. W
	frege78.a	|- A e. B
Assertion	frege78	|- ( R hereditary A -> ( A. a ( X R a -> a e. A ) -> ( X ( t+ ` R ) Y -> Y e. A ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege78.x	|- X e. U
2		frege78.y	|- Y e. V
3		frege78.r	|- R e. W
4		frege78.a	|- A e. B
5	1 2 3 4	frege77	|- ( X ( t+ ` R ) Y -> ( R hereditary A -> ( A. a ( X R a -> a e. A ) -> Y e. A ) ) )
6		frege17	|- ( ( X ( t+ ` R ) Y -> ( R hereditary A -> ( A. a ( X R a -> a e. A ) -> Y e. A ) ) ) -> ( R hereditary A -> ( A. a ( X R a -> a e. A ) -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) )
7	5 6	ax-mp	|- ( R hereditary A -> ( A. a ( X R a -> a e. A ) -> ( X ( t+ ` R ) Y -> Y e. A ) ) )