Theorem frege74

Description: If X has a property A that is hereditary in the R -sequence, then every result of a application of the procedure R to X has the property A . Proposition 74 of Frege1879 p. 60. (Contributed by RP, 28-Mar-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege74.x	⊢ 𝑋 ∈ 𝑈
	frege74.y	⊢ 𝑌 ∈ 𝑉
Assertion	frege74	⊢ ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege74.x	⊢ 𝑋 ∈ 𝑈
2		frege74.y	⊢ 𝑌 ∈ 𝑉
3	1 2	frege72	⊢ ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) )
4		ax-frege8	⊢ ( ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) ) → ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) ) )
5	3 4	ax-mp	⊢ ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) )