Description: Proof of a single axiom that can replace both ax-c7 and ax-11 . See axc711toc7 and axc711to11 for the rederivation of those axioms. (Contributed by NM, 18-Nov-2006) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | axc711 | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑦 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-11 | ⊢ ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 ) | |
| 2 | 1 | con3i | ⊢ ( ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ¬ ∀ 𝑦 ∀ 𝑥 𝜑 ) |
| 3 | 2 | alimi | ⊢ ( ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑥 ¬ ∀ 𝑦 ∀ 𝑥 𝜑 ) |
| 4 | 3 | con3i | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑦 ∀ 𝑥 𝜑 → ¬ ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 ) |
| 5 | ax-c7 | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 𝜑 ) | |
| 6 | 4 5 | syl | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑦 𝜑 ) |