Theorem bj-eximcom

Description: A commuted form of exim which is sometimes posited as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023)

	Ref	Expression
Assertion	bj-eximcom	$$⊢\exists x\left(\phi \to \psi \right)\to \left(\forall x\phi \to \exists x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm2.27	$$⊢\phi \to \left(\left(\phi \to \psi \right)\to \psi \right)$$
2	1	aleximi	$$⊢\forall x\phi \to \left(\exists x\left(\phi \to \psi \right)\to \exists x\psi \right)$$
3	2	com12	$$⊢\exists x\left(\phi \to \psi \right)\to \left(\forall x\phi \to \exists x\psi \right)$$