Theorem frege70

Description: Lemma for frege72 . Proposition 70 of Frege1879 p. 58. (Contributed by RP, 28-Mar-2020) (Revised by RP, 3-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege70.x	$$⊢X\in V$$
Assertion	frege70	$$⊢RhereditaryA\to \left(X\in A\to \forall y\left(XRy\to y\in A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege70.x	$$⊢X\in V$$
2		dffrege69	$$⊢\forall x\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)\leftrightarrow RhereditaryA$$
3	1	frege68c	$$⊢\left(\forall x\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)\leftrightarrow RhereditaryA\right)\to \left(RhereditaryA\to \underset{\dot{}}{[}X/x\underset{\dot{}}{]}\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)\right)$$
4		sbcel1v	$$⊢\underset{\dot{}}{[}X/x\underset{\dot{}}{]}x\in A\leftrightarrow X\in A$$
5	4	biimpri	$$⊢X\in A\to \underset{\dot{}}{[}X/x\underset{\dot{}}{]}x\in A$$
6		sbcim1	$$⊢\underset{\dot{}}{[}X/x\underset{\dot{}}{]}\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)\to \left(\underset{\dot{}}{[}X/x\underset{\dot{}}{]}x\in A\to \underset{\dot{}}{[}X/x\underset{\dot{}}{]}\forall y\left(xRy\to y\in A\right)\right)$$
7		sbcal	$$⊢\underset{\dot{}}{[}X/x\underset{\dot{}}{]}\forall y\left(xRy\to y\in A\right)\leftrightarrow \forall y\underset{\dot{}}{[}X/x\underset{\dot{}}{]}\left(xRy\to y\in A\right)$$
8		sbcim1	$$⊢\underset{\dot{}}{[}X/x\underset{\dot{}}{]}\left(xRy\to y\in A\right)\to \left(\underset{\dot{}}{[}X/x\underset{\dot{}}{]}xRy\to \underset{\dot{}}{[}X/x\underset{\dot{}}{]}y\in A\right)$$
9		sbcbr1g	$$⊢X\in V\to \left(\underset{\dot{}}{[}X/x\underset{\dot{}}{]}xRy\leftrightarrow ⦋X/x⦌xRy\right)$$
10	1 9	ax-mp	$$⊢\underset{\dot{}}{[}X/x\underset{\dot{}}{]}xRy\leftrightarrow ⦋X/x⦌xRy$$
11		csbvarg	$$⊢X\in V\to ⦋X/x⦌x=X$$
12	1 11	ax-mp	$$⊢⦋X/x⦌x=X$$
13	12	breq1i	$$⊢⦋X/x⦌xRy\leftrightarrow XRy$$
14	10 13	bitri	$$⊢\underset{\dot{}}{[}X/x\underset{\dot{}}{]}xRy\leftrightarrow XRy$$
15		sbcg	$$⊢X\in V\to \left(\underset{\dot{}}{[}X/x\underset{\dot{}}{]}y\in A\leftrightarrow y\in A\right)$$
16	1 15	ax-mp	$$⊢\underset{\dot{}}{[}X/x\underset{\dot{}}{]}y\in A\leftrightarrow y\in A$$
17	8 14 16	3imtr3g	$$⊢\underset{\dot{}}{[}X/x\underset{\dot{}}{]}\left(xRy\to y\in A\right)\to \left(XRy\to y\in A\right)$$
18	17	alimi	$$⊢\forall y\underset{\dot{}}{[}X/x\underset{\dot{}}{]}\left(xRy\to y\in A\right)\to \forall y\left(XRy\to y\in A\right)$$
19	7 18	sylbi	$$⊢\underset{\dot{}}{[}X/x\underset{\dot{}}{]}\forall y\left(xRy\to y\in A\right)\to \forall y\left(XRy\to y\in A\right)$$
20	5 6 19	syl56	$$⊢\underset{\dot{}}{[}X/x\underset{\dot{}}{]}\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)\to \left(X\in A\to \forall y\left(XRy\to y\in A\right)\right)$$
21	3 20	syl6	$$⊢\left(\forall x\left(x\in A\to \forall y\left(xRy\to y\in A\right)\right)\leftrightarrow RhereditaryA\right)\to \left(RhereditaryA\to \left(X\in A\to \forall y\left(XRy\to y\in A\right)\right)\right)$$
22	2 21	ax-mp	$$⊢RhereditaryA\to \left(X\in A\to \forall y\left(XRy\to y\in A\right)\right)$$