Theorem frege87

Description: If Z is a result of an application of the procedure R to an object Y that follows X in the R -sequence and if every result of an application of the procedure R to X has a property A that is hereditary in the R -sequence, then Z has property A . Proposition 87 of Frege1879 p. 66. (Contributed by RP, 1-Jul-2020) (Revised by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege87.x	$$⊢X\in U$$
	frege87.y	$$⊢Y\in V$$
	frege87.z	$$⊢Z\in W$$
	frege87.r	$$⊢R\in S$$
	frege87.a	$$⊢A\in B$$
Assertion	frege87	$$⊢X\mathrm{t+}\left(R\right)Y\to \left(\forall w\left(XRw\to w\in A\right)\to \left(RhereditaryA\to \left(YRZ\to Z\in A\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege87.x	$$⊢X\in U$$
2		frege87.y	$$⊢Y\in V$$
3		frege87.z	$$⊢Z\in W$$
4		frege87.r	$$⊢R\in S$$
5		frege87.a	$$⊢A\in B$$
6	2 3	frege73	$$⊢\left(RhereditaryA\to Y\in A\right)\to \left(RhereditaryA\to \left(YRZ\to Z\in A\right)\right)$$
7	1 2 4 5	frege86	$$⊢\left(\left(RhereditaryA\to Y\in A\right)\to \left(RhereditaryA\to \left(YRZ\to Z\in A\right)\right)\right)\to \left(X\mathrm{t+}\left(R\right)Y\to \left(\forall w\left(XRw\to w\in A\right)\to \left(RhereditaryA\to \left(YRZ\to Z\in A\right)\right)\right)\right)$$
8	6 7	ax-mp	$$⊢X\mathrm{t+}\left(R\right)Y\to \left(\forall w\left(XRw\to w\in A\right)\to \left(RhereditaryA\to \left(YRZ\to Z\in A\right)\right)\right)$$