Theorem frege111

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 or precedes Z in the R -sequence. Proposition 111 of Frege1879 p. 75. (Contributed by RP, 7-Jul-2020) (Revised by RP, 8-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege111.z	⊢ 𝑍 ∈ 𝐴
	frege111.y	⊢ 𝑌 ∈ 𝐵
	frege111.v	⊢ 𝑉 ∈ 𝐶
	frege111.r	⊢ 𝑅 ∈ 𝐷
Assertion	frege111	⊢ ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉 → ( ¬ 𝑉 ( t+ ‘ 𝑅 ) 𝑍 → 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege111.z	⊢ 𝑍 ∈ 𝐴
2		frege111.y	⊢ 𝑌 ∈ 𝐵
3		frege111.v	⊢ 𝑉 ∈ 𝐶
4		frege111.r	⊢ 𝑅 ∈ 𝐷
5	1 2 3 4	frege108	⊢ ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉 → 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 ) )
6		frege25	⊢ ( ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉 → 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 ) ) → ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉 → ( ¬ 𝑉 ( t+ ‘ 𝑅 ) 𝑍 → 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 ) ) ) )
7	5 6	ax-mp	⊢ ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉 → ( ¬ 𝑉 ( t+ ‘ 𝑅 ) 𝑍 → 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 ) ) )