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		ax-1	$$⊢\phi \to \left(\exists x\phi \to \phi \right)$$
2		ax-1	$$⊢\forall x\phi \to \left(\phi \to \forall x\phi \right)$$
3	1 2	anim12i	$$⊢\left(\phi \wedge \forall x\phi \right)\to \left(\left(\exists x\phi \to \phi \right)\wedge \left(\phi \to \forall x\phi \right)\right)$$
4		df-bj-nnf	$$⊢Ⅎ\text{'}x\phi \leftrightarrow \left(\left(\exists x\phi \to \phi \right)\wedge \left(\phi \to \forall x\phi \right)\right)$$
5	3 4	sylibr	$$⊢\left(\phi \wedge \forall x\phi \right)\to Ⅎ\text{'}x\phi$$