Theorem frege96

Description: Every result of an application of the procedure R to an object that follows X in the R -sequence follows X in the R -sequence. Proposition 96 of Frege1879 p. 71. (Contributed by RP, 2-Jul-2020) (Revised by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege95.x	$$⊢X\in U$$
	frege95.y	$$⊢Y\in V$$
	frege95.z	$$⊢Z\in W$$
	frege95.r	$$⊢R\in A$$
Assertion	frege96	$$⊢X\mathrm{t+}\left(R\right)Y\to \left(YRZ\to X\mathrm{t+}\left(R\right)Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege95.x	$$⊢X\in U$$
2		frege95.y	$$⊢Y\in V$$
3		frege95.z	$$⊢Z\in W$$
4		frege95.r	$$⊢R\in A$$
5	1 2 3 4	frege95	$$⊢YRZ\to \left(X\mathrm{t+}\left(R\right)Y\to X\mathrm{t+}\left(R\right)Z\right)$$
6		ax-frege8	$$⊢\left(YRZ\to \left(X\mathrm{t+}\left(R\right)Y\to X\mathrm{t+}\left(R\right)Z\right)\right)\to \left(X\mathrm{t+}\left(R\right)Y\to \left(YRZ\to X\mathrm{t+}\left(R\right)Z\right)\right)$$
7	5 6	ax-mp	$$⊢X\mathrm{t+}\left(R\right)Y\to \left(YRZ\to X\mathrm{t+}\left(R\right)Z\right)$$