Description: Rederivation of ax-c7 from axc5c711 . Note that ax-c7 and ax-11 are not used by the rederivation. The use of alimi (which uses ax-c5 ) is allowed since we have already proved axc5c711toc5 . (Contributed by NM, 19-Nov-2006) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | axc5c711toc7 | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hba1-o | ⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 ) | |
| 2 | 1 | con3i | ⊢ ( ¬ ∀ 𝑥 ∀ 𝑥 𝜑 → ¬ ∀ 𝑥 𝜑 ) |
| 3 | 2 | alimi | ⊢ ( ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 ) |
| 4 | 3 | sps-o | ⊢ ( ∀ 𝑥 ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 ) |
| 5 | 4 | con3i | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ¬ ∀ 𝑥 ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑥 𝜑 ) |
| 6 | pm2.21 | ⊢ ( ¬ ∀ 𝑥 ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑥 𝜑 → ( ∀ 𝑥 ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) ) | |
| 7 | axc5c711 | ⊢ ( ( ∀ 𝑥 ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) → 𝜑 ) | |
| 8 | 5 6 7 | 3syl | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → 𝜑 ) |