Theorem frege103

Description: Proposition 103 of Frege1879 p. 73. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege103.z	$$⊢Z\in V$$
Assertion	frege103	$$⊢\left(Z=X\to X=Z\right)\to \left(X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(\neg X\mathrm{t+}\left(R\right)Z\to X=Z\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege103.z	$$⊢Z\in V$$
2	1	frege100	$$⊢X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(\neg X\mathrm{t+}\left(R\right)Z\to Z=X\right)$$
3		frege19	$$⊢\left(X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(\neg X\mathrm{t+}\left(R\right)Z\to Z=X\right)\right)\to \left(\left(Z=X\to X=Z\right)\to \left(X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(\neg X\mathrm{t+}\left(R\right)Z\to X=Z\right)\right)\right)$$
4	2 3	ax-mp	$$⊢\left(Z=X\to X=Z\right)\to \left(X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(\neg X\mathrm{t+}\left(R\right)Z\to X=Z\right)\right)$$