Theorem bj-nnfv

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

	Ref	Expression
Assertion	bj-nnfv	$$⊢Ⅎ\text{'}x\phi$$

Proof

Step	Hyp	Ref	Expression
1		ax5e	$$⊢\exists x\phi \to \phi$$
2		ax-5	$$⊢\phi \to \forall x\phi$$
3		df-bj-nnf	$$⊢Ⅎ\text{'}x\phi \leftrightarrow \left(\left(\exists x\phi \to \phi \right)\wedge \left(\phi \to \forall x\phi \right)\right)$$
4	1 2 3	mpbir2an	$$⊢Ⅎ\text{'}x\phi$$