Theorem frege73

Description: Lemma for frege87 . Proposition 73 of Frege1879 p. 59. (Contributed by RP, 28-Mar-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege73.x	$$⊢X\in U$$
	frege73.y	$$⊢Y\in V$$
Assertion	frege73	$$⊢\left(RhereditaryA\to X\in A\right)\to \left(RhereditaryA\to \left(XRY\to Y\in A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege73.x	$$⊢X\in U$$
2		frege73.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-frege2	$$⊢\left(RhereditaryA\to \left(X\in A\to \left(XRY\to Y\in A\right)\right)\right)\to \left(\left(RhereditaryA\to X\in A\right)\to \left(RhereditaryA\to \left(XRY\to Y\in A\right)\right)\right)$$
5	3 4	ax-mp	$$⊢\left(RhereditaryA\to X\in A\right)\to \left(RhereditaryA\to \left(XRY\to Y\in A\right)\right)$$