Theorem frege85

Description: Commuted form of frege77 . Proposition 85 of Frege1879 p. 66. (Contributed by RP, 1-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege84.x	$$⊢X\in U$$
	frege84.y	$$⊢Y\in V$$
	frege84.r	$$⊢R\in W$$
	frege84.a	$$⊢A\in B$$
Assertion	frege85	$$⊢X\mathrm{t+}\left(R\right)Y\to \left(\forall z\left(XRz\to z\in A\right)\to \left(RhereditaryA\to Y\in A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege84.x	$$⊢X\in U$$
2		frege84.y	$$⊢Y\in V$$
3		frege84.r	$$⊢R\in W$$
4		frege84.a	$$⊢A\in B$$
5	1 2 3 4	frege77	$$⊢X\mathrm{t+}\left(R\right)Y\to \left(RhereditaryA\to \left(\forall z\left(XRz\to z\in A\right)\to Y\in A\right)\right)$$
6		frege12	$$⊢\left(X\mathrm{t+}\left(R\right)Y\to \left(RhereditaryA\to \left(\forall z\left(XRz\to z\in A\right)\to Y\in A\right)\right)\right)\to \left(X\mathrm{t+}\left(R\right)Y\to \left(\forall z\left(XRz\to z\in A\right)\to \left(RhereditaryA\to Y\in A\right)\right)\right)$$
7	5 6	ax-mp	$$⊢X\mathrm{t+}\left(R\right)Y\to \left(\forall z\left(XRz\to z\in A\right)\to \left(RhereditaryA\to Y\in A\right)\right)$$