Theorem exlimim

Description: Closed form of exlimimd . (Contributed by ML, 17-Jul-2020)

	Ref	Expression
Assertion	exlimim	$$⊢\left(\exists x\phi \wedge \forall x\left(\phi \to \psi \right)\right)\to \psi$$

Proof

Step	Hyp	Ref	Expression
1		nfa1	$$⊢Ⅎx\forall x\left(\phi \to \psi \right)$$
2		nfv	$$⊢Ⅎx\psi$$
3		sp	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\phi \to \psi \right)$$
4	1 2 3	exlimd	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \psi \right)$$
5	4	impcom	$$⊢\left(\exists x\phi \wedge \forall x\left(\phi \to \psi \right)\right)\to \psi$$