Theorem frege82

Description: Closed-form deduction based on frege81 . Proposition 82 of Frege1879 p. 64. (Contributed by RP, 1-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege82.x	⊢ 𝑋 ∈ 𝑈
	frege82.y	⊢ 𝑌 ∈ 𝑉
	frege82.r	⊢ 𝑅 ∈ 𝑊
	frege82.a	⊢ 𝐴 ∈ 𝐵
Assertion	frege82	⊢ ( ( 𝜑 → 𝑋 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝜑 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege82.x	⊢ 𝑋 ∈ 𝑈
2		frege82.y	⊢ 𝑌 ∈ 𝑉
3		frege82.r	⊢ 𝑅 ∈ 𝑊
4		frege82.a	⊢ 𝐴 ∈ 𝐵
5	1 2 3 4	frege81	⊢ ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) )
6		frege18	⊢ ( ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) ) → ( ( 𝜑 → 𝑋 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝜑 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) ) ) )
7	5 6	ax-mp	⊢ ( ( 𝜑 → 𝑋 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝜑 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) ) )