Theorem bj-nnf-alrim

Description: Proof of the closed form of alrimi from modalK (compare alrimiv ). See also bj-alrim . Actually, most proofs between 19.3t and 2sbbid could be proved without ax-12 . (Contributed by BJ, 20-Aug-2023)

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

Proof

Step	Hyp	Ref	Expression
1		bj-nnfa	$$⊢Ⅎ\text{'}x\phi \to \left(\phi \to \forall x\phi \right)$$
2		alim	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\forall x\phi \to \forall x\psi \right)$$
3	1 2	syl9	$$⊢Ⅎ\text{'}x\phi \to \left(\forall x\left(\phi \to \psi \right)\to \left(\phi \to \forall x\psi \right)\right)$$