Description: Proposition 113 of Frege1879 p. 76. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | frege112.z | ⊢ 𝑍 ∈ 𝑉 | |
| Assertion | frege113 | ⊢ ( ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 → ( ¬ 𝑍 ( t+ ‘ 𝑅 ) 𝑋 → 𝑍 = 𝑋 ) ) → ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 → ( ¬ 𝑍 ( t+ ‘ 𝑅 ) 𝑋 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frege112.z | ⊢ 𝑍 ∈ 𝑉 | |
| 2 | 1 | frege112 | ⊢ ( 𝑍 = 𝑋 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) |
| 3 | frege7 | ⊢ ( ( 𝑍 = 𝑋 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) → ( ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 → ( ¬ 𝑍 ( t+ ‘ 𝑅 ) 𝑋 → 𝑍 = 𝑋 ) ) → ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 → ( ¬ 𝑍 ( t+ ‘ 𝑅 ) 𝑋 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) ) ) ) | |
| 4 | 2 3 | ax-mp | ⊢ ( ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 → ( ¬ 𝑍 ( t+ ‘ 𝑅 ) 𝑋 → 𝑍 = 𝑋 ) ) → ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 → ( ¬ 𝑍 ( t+ ‘ 𝑅 ) 𝑋 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) ) ) |