Theorem frege87

Description: If Z is a result of an application of the procedure R to an object Y that follows X in the R -sequence and if every result of an application of the procedure R to X has a property A that is hereditary in the R -sequence, then Z has property A . Proposition 87 of Frege1879 p. 66. (Contributed by RP, 1-Jul-2020) (Revised by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege87.x	⊢ 𝑋 ∈ 𝑈
	frege87.y	⊢ 𝑌 ∈ 𝑉
	frege87.z	⊢ 𝑍 ∈ 𝑊
	frege87.r	⊢ 𝑅 ∈ 𝑆
	frege87.a	⊢ 𝐴 ∈ 𝐵
Assertion	frege87	⊢ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑌 𝑅 𝑍 → 𝑍 ∈ 𝐴 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege87.x	⊢ 𝑋 ∈ 𝑈
2		frege87.y	⊢ 𝑌 ∈ 𝑉
3		frege87.z	⊢ 𝑍 ∈ 𝑊
4		frege87.r	⊢ 𝑅 ∈ 𝑆
5		frege87.a	⊢ 𝐴 ∈ 𝐵
6	2 3	frege73	⊢ ( ( 𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑌 𝑅 𝑍 → 𝑍 ∈ 𝐴 ) ) )
7	1 2 4 5	frege86	⊢ ( ( ( 𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑌 𝑅 𝑍 → 𝑍 ∈ 𝐴 ) ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑌 𝑅 𝑍 → 𝑍 ∈ 𝐴 ) ) ) ) )
8	6 7	ax-mp	⊢ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑌 𝑅 𝑍 → 𝑍 ∈ 𝐴 ) ) ) )