Theorem frege80

Description: Add additional condition to both clauses of frege79 . Proposition 80 of Frege1879 p. 63. (Contributed by RP, 1-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege80.x	|- X e. U
	frege80.y	|- Y e. V
	frege80.r	|- R e. W
	frege80.a	|- A e. B
Assertion	frege80	|- ( ( X e. A -> ( R hereditary A -> A. a ( X R a -> a e. A ) ) ) -> ( X e. A -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege80.x	|- X e. U
2		frege80.y	|- Y e. V
3		frege80.r	|- R e. W
4		frege80.a	|- A e. B
5	1 2 3 4	frege79	|- ( ( R hereditary A -> A. a ( X R a -> a e. A ) ) -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) )
6		frege5	|- ( ( ( R hereditary A -> A. a ( X R a -> a e. A ) ) -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) -> ( ( X e. A -> ( R hereditary A -> A. a ( X R a -> a e. A ) ) ) -> ( X e. A -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) ) )
7	5 6	ax-mp	|- ( ( X e. A -> ( R hereditary A -> A. a ( X R a -> a e. A ) ) ) -> ( X e. A -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) )