Theorem frege85

Description: Commuted form of frege77 . Proposition 85 of Frege1879 p. 66. (Contributed by RP, 1-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege84.x	⊢ 𝑋 ∈ 𝑈
	frege84.y	⊢ 𝑌 ∈ 𝑉
	frege84.r	⊢ 𝑅 ∈ 𝑊
	frege84.a	⊢ 𝐴 ∈ 𝐵
Assertion	frege85	⊢ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege84.x	⊢ 𝑋 ∈ 𝑈
2		frege84.y	⊢ 𝑌 ∈ 𝑉
3		frege84.r	⊢ 𝑅 ∈ 𝑊
4		frege84.a	⊢ 𝐴 ∈ 𝐵
5	1 2 3 4	frege77	⊢ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( 𝑅 hereditary 𝐴 → ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) → 𝑌 ∈ 𝐴 ) ) )
6		frege12	⊢ ( ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( 𝑅 hereditary 𝐴 → ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) → 𝑌 ∈ 𝐴 ) ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴 ) ) ) )
7	5 6	ax-mp	⊢ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴 ) ) )