Theorem frege88

Description: Commuted form of frege87 . Proposition 88 of Frege1879 p. 67. (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	frege88	$$⊢YRZ\to \left(X\mathrm{t+}\left(R\right)Y\to \left(\forall w\left(XRw\to w\in A\right)\to \left(RhereditaryA\to Z\in A\right)\right)\right)$$

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	1 2 3 4 5	frege87	$$⊢X\mathrm{t+}\left(R\right)Y\to \left(\forall w\left(XRw\to w\in A\right)\to \left(RhereditaryA\to \left(YRZ\to Z\in A\right)\right)\right)$$
7		frege15	$$⊢\left(X\mathrm{t+}\left(R\right)Y\to \left(\forall w\left(XRw\to w\in A\right)\to \left(RhereditaryA\to \left(YRZ\to Z\in A\right)\right)\right)\right)\to \left(YRZ\to \left(X\mathrm{t+}\left(R\right)Y\to \left(\forall w\left(XRw\to w\in A\right)\to \left(RhereditaryA\to Z\in A\right)\right)\right)\right)$$
8	6 7	ax-mp	$$⊢YRZ\to \left(X\mathrm{t+}\left(R\right)Y\to \left(\forall w\left(XRw\to w\in A\right)\to \left(RhereditaryA\to Z\in A\right)\right)\right)$$