Metamath Proof Explorer


Theorem frege107

Description: Proposition 107 of Frege1879 p. 74. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

Ref Expression
Hypothesis frege107.v 𝑉𝐴
Assertion frege107 ( ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉𝑍 ( t+ ‘ 𝑅 ) 𝑉 ) ) → ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 ) ) )

Proof

Step Hyp Ref Expression
1 frege107.v 𝑉𝐴
2 1 frege106 ( 𝑍 ( t+ ‘ 𝑅 ) 𝑉𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 )
3 frege7 ( ( 𝑍 ( t+ ‘ 𝑅 ) 𝑉𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 ) → ( ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉𝑍 ( t+ ‘ 𝑅 ) 𝑉 ) ) → ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 ) ) ) )
4 2 3 ax-mp ( ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉𝑍 ( t+ ‘ 𝑅 ) 𝑉 ) ) → ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 ) ) )