Theorem frege72

Description: If property A is hereditary in the R -sequence, if x has property A , and if y is a result of an application of the procedure R to x , then y has property A . Proposition 72 of Frege1879 p. 59. (Contributed by RP, 28-Mar-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege72.x	$$⊢X\in U$$
	frege72.y	$$⊢Y\in V$$
Assertion	frege72	$$⊢RhereditaryA\to \left(X\in A\to \left(XRY\to Y\in A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege72.x	$$⊢X\in U$$
2		frege72.y	$$⊢Y\in V$$
3	2	frege58c	$$⊢\forall z\left(XRz\to z\in A\right)\to \underset{\dot{}}{[}Y/z\underset{\dot{}}{]}\left(XRz\to z\in A\right)$$
4		sbcim1	$$⊢\underset{\dot{}}{[}Y/z\underset{\dot{}}{]}\left(XRz\to z\in A\right)\to \left(\underset{\dot{}}{[}Y/z\underset{\dot{}}{]}XRz\to \underset{\dot{}}{[}Y/z\underset{\dot{}}{]}z\in A\right)$$
5		sbcbr2g	$$⊢Y\in V\to \left(\underset{\dot{}}{[}Y/z\underset{\dot{}}{]}XRz\leftrightarrow XR⦋Y/z⦌z\right)$$
6		csbvarg	$$⊢Y\in V\to ⦋Y/z⦌z=Y$$
7	6	breq2d	$$⊢Y\in V\to \left(XR⦋Y/z⦌z\leftrightarrow XRY\right)$$
8	5 7	bitrd	$$⊢Y\in V\to \left(\underset{\dot{}}{[}Y/z\underset{\dot{}}{]}XRz\leftrightarrow XRY\right)$$
9	2 8	ax-mp	$$⊢\underset{\dot{}}{[}Y/z\underset{\dot{}}{]}XRz\leftrightarrow XRY$$
10		sbcel1v	$$⊢\underset{\dot{}}{[}Y/z\underset{\dot{}}{]}z\in A\leftrightarrow Y\in A$$
11	4 9 10	3imtr3g	$$⊢\underset{\dot{}}{[}Y/z\underset{\dot{}}{]}\left(XRz\to z\in A\right)\to \left(XRY\to Y\in A\right)$$
12	3 11	syl	$$⊢\forall z\left(XRz\to z\in A\right)\to \left(XRY\to Y\in A\right)$$
13	1	frege71	$$⊢\left(\forall z\left(XRz\to z\in A\right)\to \left(XRY\to Y\in A\right)\right)\to \left(RhereditaryA\to \left(X\in A\to \left(XRY\to Y\in A\right)\right)\right)$$
14	12 13	ax-mp	$$⊢RhereditaryA\to \left(X\in A\to \left(XRY\to Y\in A\right)\right)$$