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	$$⊢\forall x\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)\to RhereditaryA$$

Proof

Step	Hyp	Ref	Expression
1		dffrege69	$$⊢\forall x\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)\leftrightarrow RhereditaryA$$
2		frege52aid	$$⊢\left(\forall x\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)\leftrightarrow RhereditaryA\right)\to \left(\forall x\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)\to RhereditaryA\right)$$
3	1 2	ax-mp	$$⊢\forall x\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)\to RhereditaryA$$