Theorem frege108

Description: If Y belongs to the R -sequence beginning with Z , then every result of an application of the procedure R to Y belongs to the R -sequence beginning with Z . Proposition 108 of Frege1879 p. 74. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege108.z	$$⊢Z\in A$$
	frege108.y	$$⊢Y\in B$$
	frege108.v	$$⊢V\in C$$
	frege108.r	$$⊢R\in D$$
Assertion	frege108	$$⊢Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)V\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege108.z	$$⊢Z\in A$$
2		frege108.y	$$⊢Y\in B$$
3		frege108.v	$$⊢V\in C$$
4		frege108.r	$$⊢R\in D$$
5	1 2 3 4	frege102	$$⊢Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to Z\mathrm{t+}\left(R\right)V\right)$$
6	3	frege107	$$⊢\left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to Z\mathrm{t+}\left(R\right)V\right)\right)\to \left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)V\right)\right)$$
7	5 6	ax-mp	$$⊢Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)V\right)$$