Theorem frege102

Description: If Z belongs to the R -sequence beginning with X , then every result of an application of the procedure R to Z follows X in the R -sequence. Proposition 102 of Frege1879 p. 72. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege102.x	$$⊢X\in A$$
	frege102.z	$$⊢Z\in B$$
	frege102.v	$$⊢V\in C$$
	frege102.r	$$⊢R\in D$$
Assertion	frege102	$$⊢X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(ZRV\to X\mathrm{t+}\left(R\right)V\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege102.x	$$⊢X\in A$$
2		frege102.z	$$⊢Z\in B$$
3		frege102.v	$$⊢V\in C$$
4		frege102.r	$$⊢R\in D$$
5	2 3 4	frege92	$$⊢Z=X\to \left(ZRV\to X\mathrm{t+}\left(R\right)V\right)$$
6	1 2 3 4	frege96	$$⊢X\mathrm{t+}\left(R\right)Z\to \left(ZRV\to X\mathrm{t+}\left(R\right)V\right)$$
7	2	frege101	$$⊢\left(Z=X\to \left(ZRV\to X\mathrm{t+}\left(R\right)V\right)\right)\to \left(\left(X\mathrm{t+}\left(R\right)Z\to \left(ZRV\to X\mathrm{t+}\left(R\right)V\right)\right)\to \left(X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(ZRV\to X\mathrm{t+}\left(R\right)V\right)\right)\right)$$
8	5 6 7	mp2	$$⊢X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(ZRV\to X\mathrm{t+}\left(R\right)V\right)$$