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Metamath Proof Explorer
Theorem nf4
Description: Alternate definition of nonfreeness. This definition uses only primitive
symbols ( -> , -. , A. ). (Contributed by BJ, 16-Sep-2021)
|
|
Ref |
Expression |
|
Assertion |
nf4 |
⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ 𝜑 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
nf3 |
⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∀ 𝑥 𝜑 ∨ ∀ 𝑥 ¬ 𝜑 ) ) |
| 2 |
|
df-or |
⊢ ( ( ∀ 𝑥 𝜑 ∨ ∀ 𝑥 ¬ 𝜑 ) ↔ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ 𝜑 ) ) |
| 3 |
1 2
|
bitri |
⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ 𝜑 ) ) |