Theorem bj-nnfv

Description: A non-occurring variable is nonfree in a formula. (Contributed by BJ, 28-Jul-2023)

	Ref	Expression
Assertion	bj-nnfv	⊢ Ⅎ' 𝑥 𝜑

Proof

Step	Hyp	Ref	Expression
1		ax5e	⊢ ( ∃ 𝑥 𝜑 → 𝜑 )
2		ax-5	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
3		df-bj-nnf	⊢ ( Ⅎ' 𝑥 𝜑 ↔ ( ( ∃ 𝑥 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑥 𝜑 ) ) )
4	1 2 3	mpbir2an	⊢ Ⅎ' 𝑥 𝜑