Theorem frege75

Description: If from the proposition that x has property A , whatever x may be, it can be inferred that every result of an application of the procedure R to x has property A , then property A is hereditary in the R -sequence. Proposition 75 of Frege1879 p. 60. (Contributed by RP, 28-Mar-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege75	|- ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) -> R hereditary A )

Proof

Step	Hyp	Ref	Expression
1		dffrege69	|- ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) <-> R hereditary A )
2		frege52aid	|- ( ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) <-> R hereditary A ) -> ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) -> R hereditary A ) )
3	1 2	ax-mp	|- ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) -> R hereditary A )