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 ∈ 𝑈
	frege95.y	⊢ 𝑌 ∈ 𝑉
	frege95.z	⊢ 𝑍 ∈ 𝑊
	frege95.r	⊢ 𝑅 ∈ 𝐴
Assertion	frege96	⊢ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( 𝑌 𝑅 𝑍 → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) )

Proof

Step	Hyp	Ref	Expression
1		frege95.x	⊢ 𝑋 ∈ 𝑈
2		frege95.y	⊢ 𝑌 ∈ 𝑉
3		frege95.z	⊢ 𝑍 ∈ 𝑊
4		frege95.r	⊢ 𝑅 ∈ 𝐴
5	1 2 3 4	frege95	⊢ ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) )
6		ax-frege8	⊢ ( ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( 𝑌 𝑅 𝑍 → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) ) )
7	5 6	ax-mp	⊢ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( 𝑌 𝑅 𝑍 → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) )