Theorem frege113

Description: Proposition 113 of Frege1879 p. 76. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege112.z	$$⊢Z\in V$$
Assertion	frege113	$$⊢\left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\to \left(\neg Z\mathrm{t+}\left(R\right)X\to Z=X\right)\right)\to \left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\to \left(\neg Z\mathrm{t+}\left(R\right)X\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege112.z	$$⊢Z\in V$$
2	1	frege112	$$⊢Z=X\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z$$
3		frege7	$$⊢\left(Z=X\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\right)\to \left(\left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\to \left(\neg Z\mathrm{t+}\left(R\right)X\to Z=X\right)\right)\to \left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\to \left(\neg Z\mathrm{t+}\left(R\right)X\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\right)\right)\right)$$
4	2 3	ax-mp	$$⊢\left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\to \left(\neg Z\mathrm{t+}\left(R\right)X\to Z=X\right)\right)\to \left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\to \left(\neg Z\mathrm{t+}\left(R\right)X\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\right)\right)$$