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Metamath Proof Explorer
Description: Alternate definition of df-bj-nnf using only primitive symbols
( -> , -. , A. ) in each conjunct. (Contributed by BJ, 20-Aug-2023)
|
|
Ref |
Expression |
|
Assertion |
bj-dfnnf2 |
⊢ ( Ⅎ' 𝑥 𝜑 ↔ ( ( 𝜑 → ∀ 𝑥 𝜑 ) ∧ ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
df-bj-nnf |
⊢ ( Ⅎ' 𝑥 𝜑 ↔ ( ( ∃ 𝑥 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑥 𝜑 ) ) ) |
| 2 |
|
eximal |
⊢ ( ( ∃ 𝑥 𝜑 → 𝜑 ) ↔ ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) ) |
| 3 |
2
|
anbi2ci |
⊢ ( ( ( ∃ 𝑥 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑥 𝜑 ) ) ↔ ( ( 𝜑 → ∀ 𝑥 𝜑 ) ∧ ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) ) ) |
| 4 |
1 3
|
bitri |
⊢ ( Ⅎ' 𝑥 𝜑 ↔ ( ( 𝜑 → ∀ 𝑥 𝜑 ) ∧ ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) ) ) |