Description: Proof of a single axiom that can replace ax-c5 , ax-c7 , and ax-11 in a subsystem that includes these axioms plus ax-c4 and ax-gen (and propositional calculus). See axc5c711toc5 , axc5c711toc7 , and axc5c711to11 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 . (Contributed by NM, 18-Nov-2006) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | axc5c711 | ⊢ ( ( ∀ 𝑥 ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑥 𝜑 ) → 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-c5 | ⊢ ( ∀ 𝑦 𝜑 → 𝜑 ) | |
| 2 | ax10fromc7 | ⊢ ( ¬ ∀ 𝑦 𝜑 → ∀ 𝑦 ¬ ∀ 𝑦 𝜑 ) | |
| 3 | ax-c7 | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 𝜑 ) | |
| 4 | 3 | con1i | ⊢ ( ¬ ∀ 𝑦 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 ) |
| 5 | 4 | alimi | ⊢ ( ∀ 𝑦 ¬ ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 ) |
| 6 | ax-11 | ⊢ ( ∀ 𝑦 ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 ) | |
| 7 | 2 5 6 | 3syl | ⊢ ( ¬ ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 ) |
| 8 | 1 7 | nsyl4 | ⊢ ( ¬ ∀ 𝑥 ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → 𝜑 ) |
| 9 | ax-c5 | ⊢ ( ∀ 𝑥 𝜑 → 𝜑 ) | |
| 10 | 8 9 | ja | ⊢ ( ( ∀ 𝑥 ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑥 𝜑 ) → 𝜑 ) |