Theorem frege78

Description: Commuted form of frege77 . Proposition 78 of Frege1879 p. 63. (Contributed by RP, 1-Jul-2020) (Revised by RP, 2-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege78.x	$$⊢X\in U$$
	frege78.y	$$⊢Y\in V$$
	frege78.r	$$⊢R\in W$$
	frege78.a	$$⊢A\in B$$
Assertion	frege78	$$⊢RhereditaryA\to \left(\forall a\left(XRa\to a\in A\right)\to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege78.x	$$⊢X\in U$$
2		frege78.y	$$⊢Y\in V$$
3		frege78.r	$$⊢R\in W$$
4		frege78.a	$$⊢A\in B$$
5	1 2 3 4	frege77	$$⊢X\mathrm{t+}\left(R\right)Y\to \left(RhereditaryA\to \left(\forall a\left(XRa\to a\in A\right)\to Y\in A\right)\right)$$
6		frege17	$$⊢\left(X\mathrm{t+}\left(R\right)Y\to \left(RhereditaryA\to \left(\forall a\left(XRa\to a\in A\right)\to Y\in A\right)\right)\right)\to \left(RhereditaryA\to \left(\forall a\left(XRa\to a\in A\right)\to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)\right)$$
7	5 6	ax-mp	$$⊢RhereditaryA\to \left(\forall a\left(XRa\to a\in A\right)\to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)$$