Theorem frege114

Description: If X belongs to the R -sequence beginning with Z , then Z belongs to the R -sequence beginning with X or X follows Z in the R -sequence. Proposition 114 of Frege1879 p. 76. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege114.x	$$⊢X\in U$$
	frege114.z	$$⊢Z\in V$$
Assertion	frege114	$$⊢Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\to \left(\neg Z\mathrm{t+}\left(R\right)X\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege114.x	$$⊢X\in U$$
2		frege114.z	$$⊢Z\in V$$
3	1	frege104	$$⊢Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\to \left(\neg Z\mathrm{t+}\left(R\right)X\to Z=X\right)$$
4	2	frege113	$$⊢\left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\to \left(\neg Z\mathrm{t+}\left(R\right)X\to Z=X\right)\right)\to \left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\to \left(\neg Z\mathrm{t+}\left(R\right)X\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\right)\right)$$
5	3 4	ax-mp	$$⊢Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\to \left(\neg Z\mathrm{t+}\left(R\right)X\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\right)$$