Theorem dffrege69

Description: If from the proposition that x has property A it can be inferred generally, whatever x may be, that every result of an application of the procedure R to x has property A , then we say " Property A is hereditary in the R -sequence. Definition 69 of Frege1879 p. 55. (Contributed by RP, 28-Mar-2020)

	Ref	Expression
Assertion	dffrege69	$$⊢\forall x\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)\leftrightarrow RhereditaryA$$

Proof

Step	Hyp	Ref	Expression
1		dfhe3	$$⊢RhereditaryA\leftrightarrow \forall x\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)$$
2	1	bicomi	$$⊢\forall x\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)\leftrightarrow RhereditaryA$$