Theorem bj-nnftht

Description: A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp (modal T), as in bj-nnfbi . (Contributed by BJ, 28-Jul-2023)

	Ref	Expression
Assertion	bj-nnftht	$$⊢\left(\phi \wedge \forall x\phi \right)\to Ⅎ\text{'}x\phi$$

Proof

Step	Hyp	Ref	Expression
1		bj-alnnf2	$$⊢\phi \to \left(\forall x\phi \leftrightarrow Ⅎ\text{'}x\phi \right)$$
2	1	biimpa	$$⊢\left(\phi \wedge \forall x\phi \right)\to Ⅎ\text{'}x\phi$$