Theorem frege71

Description: Lemma for frege72 . Proposition 71 of Frege1879 p. 59. (Contributed by RP, 28-Mar-2020) (Revised by RP, 3-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege71.x	⊢ 𝑋 ∈ 𝑉
Assertion	frege71	⊢ ( ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) → ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege71.x	⊢ 𝑋 ∈ 𝑉
2	1	frege70	⊢ ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) ) )
3		frege19	⊢ ( ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) ) ) → ( ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) → ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) ) ) )
4	2 3	ax-mp	⊢ ( ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) → ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) ) )