Theorem frege78

Description: Commuted form of frege77 . Proposition 78 of Frege1879 p. 63. (Contributed by RP, 1-Jul-2020) (Revised by RP, 2-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege78.x	⊢ 𝑋 ∈ 𝑈
	frege78.y	⊢ 𝑌 ∈ 𝑉
	frege78.r	⊢ 𝑅 ∈ 𝑊
	frege78.a	⊢ 𝐴 ∈ 𝐵
Assertion	frege78	⊢ ( 𝑅 hereditary 𝐴 → ( ∀ 𝑎 ( 𝑋 𝑅 𝑎 → 𝑎 ∈ 𝐴 ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege78.x	⊢ 𝑋 ∈ 𝑈
2		frege78.y	⊢ 𝑌 ∈ 𝑉
3		frege78.r	⊢ 𝑅 ∈ 𝑊
4		frege78.a	⊢ 𝐴 ∈ 𝐵
5	1 2 3 4	frege77	⊢ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( 𝑅 hereditary 𝐴 → ( ∀ 𝑎 ( 𝑋 𝑅 𝑎 → 𝑎 ∈ 𝐴 ) → 𝑌 ∈ 𝐴 ) ) )
6		frege17	⊢ ( ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( 𝑅 hereditary 𝐴 → ( ∀ 𝑎 ( 𝑋 𝑅 𝑎 → 𝑎 ∈ 𝐴 ) → 𝑌 ∈ 𝐴 ) ) ) → ( 𝑅 hereditary 𝐴 → ( ∀ 𝑎 ( 𝑋 𝑅 𝑎 → 𝑎 ∈ 𝐴 ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) ) )
7	5 6	ax-mp	⊢ ( 𝑅 hereditary 𝐴 → ( ∀ 𝑎 ( 𝑋 𝑅 𝑎 → 𝑎 ∈ 𝐴 ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) )