Theorem frege95

Description: Looking one past a pair related by transitive closure of a relation. Proposition 95 of Frege1879 p. 70. (Contributed by RP, 2-Jul-2020) (Revised by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege95.x	⊢ 𝑋 ∈ 𝑈
	frege95.y	⊢ 𝑌 ∈ 𝑉
	frege95.z	⊢ 𝑍 ∈ 𝑊
	frege95.r	⊢ 𝑅 ∈ 𝐴
Assertion	frege95	⊢ ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) )

Proof

Step	Hyp	Ref	Expression
1		frege95.x	⊢ 𝑋 ∈ 𝑈
2		frege95.y	⊢ 𝑌 ∈ 𝑉
3		frege95.z	⊢ 𝑍 ∈ 𝑊
4		frege95.r	⊢ 𝑅 ∈ 𝐴
5		vex	⊢ 𝑓 ∈ V
6	1 2 3 4 5	frege88	⊢ ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑍 ∈ 𝑓 ) ) ) )
7	6	alrimdv	⊢ ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ∀ 𝑓 ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑍 ∈ 𝑓 ) ) ) )
8	1 3 4	frege94	⊢ ( ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ∀ 𝑓 ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑍 ∈ 𝑓 ) ) ) ) → ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) ) )
9	7 8	ax-mp	⊢ ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) )