Theorem frege82

Description: Closed-form deduction based on frege81 . Proposition 82 of Frege1879 p. 64. (Contributed by RP, 1-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege82.x	$$⊢X\in U$$
	frege82.y	$$⊢Y\in V$$
	frege82.r	$$⊢R\in W$$
	frege82.a	$$⊢A\in B$$
Assertion	frege82	$$⊢\left(\phi \to X\in A\right)\to \left(RhereditaryA\to \left(\phi \to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege82.x	$$⊢X\in U$$
2		frege82.y	$$⊢Y\in V$$
3		frege82.r	$$⊢R\in W$$
4		frege82.a	$$⊢A\in B$$
5	1 2 3 4	frege81	$$⊢X\in A\to \left(RhereditaryA\to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)$$
6		frege18	$$⊢\left(X\in A\to \left(RhereditaryA\to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)\right)\to \left(\left(\phi \to X\in A\right)\to \left(RhereditaryA\to \left(\phi \to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)\right)\right)$$
7	5 6	ax-mp	$$⊢\left(\phi \to X\in A\right)\to \left(RhereditaryA\to \left(\phi \to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)\right)$$