Theorem frege73

Description: Lemma for frege87 . Proposition 73 of Frege1879 p. 59. (Contributed by RP, 28-Mar-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege73.x	⊢ 𝑋 ∈ 𝑈
	frege73.y	⊢ 𝑌 ∈ 𝑉
Assertion	frege73	⊢ ( ( 𝑅 hereditary 𝐴 → 𝑋 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege73.x	⊢ 𝑋 ∈ 𝑈
2		frege73.y	⊢ 𝑌 ∈ 𝑉
3	1 2	frege72	⊢ ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) )
4		ax-frege2	⊢ ( ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) ) → ( ( 𝑅 hereditary 𝐴 → 𝑋 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) ) )
5	3 4	ax-mp	⊢ ( ( 𝑅 hereditary 𝐴 → 𝑋 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) )