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 ) ) ) |
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 ) ) ) |