Metamath Proof Explorer


Theorem frege94

Description: Looking one past a pair related by transitive closure of a relation. Proposition 94 of Frege1879 p. 70. (Contributed by RP, 2-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

Ref Expression
Hypotheses frege94.x 𝑋𝑈
frege94.z 𝑍𝑉
frege94.r 𝑅𝑊
Assertion frege94 ( ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ∀ 𝑓 ( ∀ 𝑤 ( 𝑋 𝑅 𝑤𝑤𝑓 ) → ( 𝑅 hereditary 𝑓𝑍𝑓 ) ) ) ) → ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) ) )

Proof

Step Hyp Ref Expression
1 frege94.x 𝑋𝑈
2 frege94.z 𝑍𝑉
3 frege94.r 𝑅𝑊
4 1 2 3 frege93 ( ∀ 𝑓 ( ∀ 𝑤 ( 𝑋 𝑅 𝑤𝑤𝑓 ) → ( 𝑅 hereditary 𝑓𝑍𝑓 ) ) → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 )
5 frege7 ( ( ∀ 𝑓 ( ∀ 𝑤 ( 𝑋 𝑅 𝑤𝑤𝑓 ) → ( 𝑅 hereditary 𝑓𝑍𝑓 ) ) → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) → ( ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ∀ 𝑓 ( ∀ 𝑤 ( 𝑋 𝑅 𝑤𝑤𝑓 ) → ( 𝑅 hereditary 𝑓𝑍𝑓 ) ) ) ) → ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) ) ) )
6 4 5 ax-mp ( ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ∀ 𝑓 ( ∀ 𝑤 ( 𝑋 𝑅 𝑤𝑤𝑓 ) → ( 𝑅 hereditary 𝑓𝑍𝑓 ) ) ) ) → ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) ) )