Theorem bj-nnfa1

Description: See nfa1 . (Contributed by BJ, 12-Aug-2023) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-nnfa1	⊢ Ⅎ' 𝑥 ∀ 𝑥 𝜑

Proof

Step	Hyp	Ref	Expression
1		hbe1a	⊢ ( ∃ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 )
2		bj-modal4	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 )
3		df-bj-nnf	⊢ ( Ⅎ' 𝑥 ∀ 𝑥 𝜑 ↔ ( ( ∃ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) ∧ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 ) ) )
4	1 2 3	mpbir2an	⊢ Ⅎ' 𝑥 ∀ 𝑥 𝜑