Description: Proposition 103 of Frege1879 p. 73. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | frege103.z | ⊢ 𝑍 ∈ 𝑉 | |
Assertion | frege103 | ⊢ ( ( 𝑍 = 𝑋 → 𝑋 = 𝑍 ) → ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 → ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑋 = 𝑍 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege103.z | ⊢ 𝑍 ∈ 𝑉 | |
2 | 1 | frege100 | ⊢ ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 → ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑍 = 𝑋 ) ) |
3 | frege19 | ⊢ ( ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 → ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑍 = 𝑋 ) ) → ( ( 𝑍 = 𝑋 → 𝑋 = 𝑍 ) → ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 → ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑋 = 𝑍 ) ) ) ) | |
4 | 2 3 | ax-mp | ⊢ ( ( 𝑍 = 𝑋 → 𝑋 = 𝑍 ) → ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 → ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑋 = 𝑍 ) ) ) |