Theorem bj-aleximiALT

Description: Alternate proof of aleximi from exim , which is sometimes used as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bj-aleximiALT.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
Assertion	bj-aleximiALT	$$⊢\forall x\phi \to \left(\exists x\psi \to \exists x\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-aleximiALT.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2	1	alimi	$$⊢\forall x\phi \to \forall x\left(\psi \to \chi \right)$$
3		bj-eximALT	$$⊢\forall x\left(\psi \to \chi \right)\to \left(\exists x\psi \to \exists x\chi \right)$$
4	2 3	syl	$$⊢\forall x\phi \to \left(\exists x\psi \to \exists x\chi \right)$$