Theorem frege119

Description: Lemma for frege120 . Proposition 119 of Frege1879 p. 78. (Contributed by RP, 8-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege116.x	$$⊢X\in U$$
	frege118.y	$$⊢Y\in V$$
Assertion	frege119	$$⊢\left(\forall a\left(YRa\to a=X\right)\to \left(YRA\to A=X\right)\right)\to \left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(YRA\to A=X\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege116.x	$$⊢X\in U$$
2		frege118.y	$$⊢Y\in V$$
3	1 2	frege118	$$⊢\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \forall a\left(YRa\to a=X\right)\right)$$
4		frege19	$$⊢\left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \forall a\left(YRa\to a=X\right)\right)\right)\to \left(\left(\forall a\left(YRa\to a=X\right)\to \left(YRA\to A=X\right)\right)\to \left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(YRA\to A=X\right)\right)\right)\right)$$
5	3 4	ax-mp	$$⊢\left(\forall a\left(YRa\to a=X\right)\to \left(YRA\to A=X\right)\right)\to \left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(YRA\to A=X\right)\right)\right)$$