Theorem dffrege69

Description: If from the proposition that x has property A it can be inferred generally, whatever x may be, that every result of an application of the procedure R to x has property A , then we say " Property A is hereditary in the R -sequence. Definition 69 of Frege1879 p. 55. (Contributed by RP, 28-Mar-2020)

	Ref	Expression
Assertion	dffrege69	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) ↔ 𝑅 hereditary 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		dfhe3	⊢ ( 𝑅 hereditary 𝐴 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) )
2	1	bicomi	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) ↔ 𝑅 hereditary 𝐴 )