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 ∈ 𝐴
	frege102.z	⊢ 𝑍 ∈ 𝐵
	frege102.v	⊢ 𝑉 ∈ 𝐶
	frege102.r	⊢ 𝑅 ∈ 𝐷
Assertion	frege102	⊢ ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		frege102.x	⊢ 𝑋 ∈ 𝐴
2		frege102.z	⊢ 𝑍 ∈ 𝐵
3		frege102.v	⊢ 𝑉 ∈ 𝐶
4		frege102.r	⊢ 𝑅 ∈ 𝐷
5	2 3 4	frege92	⊢ ( 𝑍 = 𝑋 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) )
6	1 2 3 4	frege96	⊢ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) )
7	2	frege101	⊢ ( ( 𝑍 = 𝑋 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) ) → ( ( 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) ) → ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) ) ) )
8	5 6 7	mp2	⊢ ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) )