Theorem frege75

Description: If from the proposition that x has property A , whatever x may be, it can be inferred that every result of an application of the procedure R to x has property A , then property A is hereditary in the R -sequence. Proposition 75 of Frege1879 p. 60. (Contributed by RP, 28-Mar-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege75	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) → 𝑅 hereditary 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		dffrege69	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) ↔ 𝑅 hereditary 𝐴 )
2		frege52aid	⊢ ( ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) ↔ 𝑅 hereditary 𝐴 ) → ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) → 𝑅 hereditary 𝐴 ) )
3	1 2	ax-mp	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) → 𝑅 hereditary 𝐴 )