Theorem frege80

Description: Add additional condition to both clauses of frege79 . Proposition 80 of Frege1879 p. 63. (Contributed by RP, 1-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege80.x	$$⊢X\in U$$
	frege80.y	$$⊢Y\in V$$
	frege80.r	$$⊢R\in W$$
	frege80.a	$$⊢A\in B$$
Assertion	frege80	$$⊢\left(X\in A\to \left(RhereditaryA\to \forall a\left(XRa\to a\in A\right)\right)\right)\to \left(X\in A\to \left(RhereditaryA\to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege80.x	$$⊢X\in U$$
2		frege80.y	$$⊢Y\in V$$
3		frege80.r	$$⊢R\in W$$
4		frege80.a	$$⊢A\in B$$
5	1 2 3 4	frege79	$$⊢\left(RhereditaryA\to \forall a\left(XRa\to a\in A\right)\right)\to \left(RhereditaryA\to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)$$
6		frege5	$$⊢\left(\left(RhereditaryA\to \forall a\left(XRa\to a\in A\right)\right)\to \left(RhereditaryA\to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)\right)\to \left(\left(X\in A\to \left(RhereditaryA\to \forall a\left(XRa\to a\in A\right)\right)\right)\to \left(X\in A\to \left(RhereditaryA\to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)\right)\right)$$
7	5 6	ax-mp	$$⊢\left(X\in A\to \left(RhereditaryA\to \forall a\left(XRa\to a\in A\right)\right)\right)\to \left(X\in A\to \left(RhereditaryA\to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)\right)$$