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 ∈ 𝐴
	frege108.y	⊢ 𝑌 ∈ 𝐵
	frege108.v	⊢ 𝑉 ∈ 𝐶
	frege108.r	⊢ 𝑅 ∈ 𝐷
Assertion	frege108	⊢ ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉 → 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		frege108.z	⊢ 𝑍 ∈ 𝐴
2		frege108.y	⊢ 𝑌 ∈ 𝐵
3		frege108.v	⊢ 𝑉 ∈ 𝐶
4		frege108.r	⊢ 𝑅 ∈ 𝐷
5	1 2 3 4	frege102	⊢ ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉 → 𝑍 ( t+ ‘ 𝑅 ) 𝑉 ) )
6	3	frege107	⊢ ( ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉 → 𝑍 ( t+ ‘ 𝑅 ) 𝑉 ) ) → ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉 → 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 ) ) )
7	5 6	ax-mp	⊢ ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉 → 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 ) )