Theorem frege107

Description: Proposition 107 of Frege1879 p. 74. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege107.v	$$⊢V\in A$$
Assertion	frege107	$$⊢\left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to Z\mathrm{t+}\left(R\right)V\right)\right)\to \left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)V\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege107.v	$$⊢V\in A$$
2	1	frege106	$$⊢Z\mathrm{t+}\left(R\right)V\to Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)V$$
3		frege7	$$⊢\left(Z\mathrm{t+}\left(R\right)V\to Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)V\right)\to \left(\left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to Z\mathrm{t+}\left(R\right)V\right)\right)\to \left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)V\right)\right)\right)$$
4	2 3	ax-mp	$$⊢\left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to Z\mathrm{t+}\left(R\right)V\right)\right)\to \left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)V\right)\right)$$