Theorem bj-dfnnf2

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 ) ) )