Theorem frege107

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

	Ref	Expression
Hypothesis	frege107.v	⊢ 𝑉 ∈ 𝐴
Assertion	frege107	⊢ ( ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉 → 𝑍 ( t+ ‘ 𝑅 ) 𝑉 ) ) → ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑉 → 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑉 ) ) )

Proof

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