Description: Rederivation of ax-11 from axc711 . Note that ax-c7 and ax-11 are not used by the rederivation. (Contributed by NM, 18-Nov-2006) (Proof modification is discouraged.) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | axc711to11 | ⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axc711toc7 | ⊢ ( ¬ ∀ 𝑦 ¬ ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ¬ ∀ 𝑥 ∀ 𝑦 𝜑 ) | |
2 | 1 | con4i | ⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ¬ ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 ) |
3 | axc711 | ⊢ ( ¬ ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑥 𝜑 ) | |
4 | 3 | alimi | ⊢ ( ∀ 𝑦 ¬ ∀ 𝑦 ¬ ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) |
5 | 2 4 | syl | ⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) |