Theorem frege80

Description: Add additional condition to both clauses of frege79 . Proposition 80 of Frege1879 p. 63. (Contributed by RP, 1-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege80.x	⊢ 𝑋 ∈ 𝑈
	frege80.y	⊢ 𝑌 ∈ 𝑉
	frege80.r	⊢ 𝑅 ∈ 𝑊
	frege80.a	⊢ 𝐴 ∈ 𝐵
Assertion	frege80	⊢ ( ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ∀ 𝑎 ( 𝑋 𝑅 𝑎 → 𝑎 ∈ 𝐴 ) ) ) → ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege80.x	⊢ 𝑋 ∈ 𝑈
2		frege80.y	⊢ 𝑌 ∈ 𝑉
3		frege80.r	⊢ 𝑅 ∈ 𝑊
4		frege80.a	⊢ 𝐴 ∈ 𝐵
5	1 2 3 4	frege79	⊢ ( ( 𝑅 hereditary 𝐴 → ∀ 𝑎 ( 𝑋 𝑅 𝑎 → 𝑎 ∈ 𝐴 ) ) → ( 𝑅 hereditary 𝐴 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) )
6		frege5	⊢ ( ( ( 𝑅 hereditary 𝐴 → ∀ 𝑎 ( 𝑋 𝑅 𝑎 → 𝑎 ∈ 𝐴 ) ) → ( 𝑅 hereditary 𝐴 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) ) → ( ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ∀ 𝑎 ( 𝑋 𝑅 𝑎 → 𝑎 ∈ 𝐴 ) ) ) → ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) ) ) )
7	5 6	ax-mp	⊢ ( ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ∀ 𝑎 ( 𝑋 𝑅 𝑎 → 𝑎 ∈ 𝐴 ) ) ) → ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) ) )