Description: A kind of Aristotelian inference. Proposition 59 of Frege1879 p. 51.
Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collectionFrom Frege to Goedel, this proof has the frege12 incorrectly referenced where frege30 is in the original. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | frege59c.a | ⊢ 𝐴 ∈ 𝐵 | |
| Assertion | frege59c | ⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → ( ¬ [ 𝐴 / 𝑥 ] 𝜓 → ¬ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frege59c.a | ⊢ 𝐴 ∈ 𝐵 | |
| 2 | 1 | frege58c | ⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) ) |
| 3 | sbcim1 | ⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ) | |
| 4 | 2 3 | syl | ⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ) |
| 5 | frege30 | ⊢ ( ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ) → ( [ 𝐴 / 𝑥 ] 𝜑 → ( ¬ [ 𝐴 / 𝑥 ] 𝜓 → ¬ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) ) ) | |
| 6 | 4 5 | ax-mp | ⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → ( ¬ [ 𝐴 / 𝑥 ] 𝜓 → ¬ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) ) |