Theorem bj-nnf-exlim

Description: Proof of the closed form of exlimi from modalK (compare exlimiv ). See also bj-sylget2 . (Contributed by BJ, 2-Dec-2023)

	Ref	Expression
Assertion	bj-nnf-exlim	$$⊢Ⅎ\text{'}x\psi \to \left(\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		exim	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \exists x\psi \right)$$
2		bj-nnfe	$$⊢Ⅎ\text{'}x\psi \to \left(\exists x\psi \to \psi \right)$$
3	1 2	syl9r	$$⊢Ⅎ\text{'}x\psi \to \left(\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \psi \right)\right)$$