Description: Rederivation of axc4 from axc5c4c711 . Note that only propositional calculus is required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | axc5c4c711toc4 | ⊢ ( ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-1 | ⊢ ( ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) → ( 𝜑 → ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) ) ) | |
| 2 | ax-1 | ⊢ ( ( 𝜑 → ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) ) → ( ∀ 𝑥 ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) → ( 𝜑 → ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) ) ) ) | |
| 3 | axc5c4c711 | ⊢ ( ( ∀ 𝑥 ∀ 𝑥 ¬ ∀ 𝑥 ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) → ( 𝜑 → ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) | |
| 4 | 1 2 3 | 3syl | ⊢ ( ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) |