Theorem frege74

Description: If X has a property A that is hereditary in the R -sequence, then every result of a application of the procedure R to X has the property A . Proposition 74 of Frege1879 p. 60. (Contributed by RP, 28-Mar-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege74.x	$$⊢X\in U$$
	frege74.y	$$⊢Y\in V$$
Assertion	frege74	$$⊢X\in A\to \left(RhereditaryA\to \left(XRY\to Y\in A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege74.x	$$⊢X\in U$$
2		frege74.y	$$⊢Y\in V$$
3	1 2	frege72	$$⊢RhereditaryA\to \left(X\in A\to \left(XRY\to Y\in A\right)\right)$$
4		ax-frege8	$$⊢\left(RhereditaryA\to \left(X\in A\to \left(XRY\to Y\in A\right)\right)\right)\to \left(X\in A\to \left(RhereditaryA\to \left(XRY\to Y\in A\right)\right)\right)$$
5	3 4	ax-mp	$$⊢X\in A\to \left(RhereditaryA\to \left(XRY\to Y\in A\right)\right)$$