Theorem frege120

Description: Simplified application of one direction of dffrege115 . Proposition 120 of Frege1879 p. 78. (Contributed by RP, 8-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege116.x	$$⊢X\in U$$
	frege118.y	$$⊢Y\in V$$
	frege120.a	$$⊢A\in W$$
Assertion	frege120	$$⊢\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(YRA\to A=X\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege116.x	$$⊢X\in U$$
2		frege118.y	$$⊢Y\in V$$
3		frege120.a	$$⊢A\in W$$
4	3	frege58c	$$⊢\forall a\left(YRa\to a=X\right)\to \underset{\dot{}}{[}A/a\underset{\dot{}}{]}\left(YRa\to a=X\right)$$
5		sbcim1	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\left(YRa\to a=X\right)\to \left(\underset{\dot{}}{[}A/a\underset{\dot{}}{]}YRa\to \underset{\dot{}}{[}A/a\underset{\dot{}}{]}a=X\right)$$
6		sbcbr2g	$$⊢A\in W\to \left(\underset{\dot{}}{[}A/a\underset{\dot{}}{]}YRa\leftrightarrow YR⦋A/a⦌a\right)$$
7	3 6	ax-mp	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}YRa\leftrightarrow YR⦋A/a⦌a$$
8		csbvarg	$$⊢A\in W\to ⦋A/a⦌a=A$$
9	3 8	ax-mp	$$⊢⦋A/a⦌a=A$$
10	9	breq2i	$$⊢YR⦋A/a⦌a\leftrightarrow YRA$$
11	7 10	bitri	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}YRa\leftrightarrow YRA$$
12		sbceq1g	$$⊢A\in W\to \left(\underset{\dot{}}{[}A/a\underset{\dot{}}{]}a=X\leftrightarrow ⦋A/a⦌a=X\right)$$
13	3 12	ax-mp	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}a=X\leftrightarrow ⦋A/a⦌a=X$$
14	9	eqeq1i	$$⊢⦋A/a⦌a=X\leftrightarrow A=X$$
15	13 14	bitri	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}a=X\leftrightarrow A=X$$
16	5 11 15	3imtr3g	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\left(YRa\to a=X\right)\to \left(YRA\to A=X\right)$$
17	4 16	syl	$$⊢\forall a\left(YRa\to a=X\right)\to \left(YRA\to A=X\right)$$
18	1 2	frege119	$$⊢\left(\forall a\left(YRa\to a=X\right)\to \left(YRA\to A=X\right)\right)\to \left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(YRA\to A=X\right)\right)\right)$$
19	17 18	ax-mp	$$⊢\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(YRA\to A=X\right)\right)$$