Theorem exellim

Description: Closed form of exellimddv . See also exlimim for a more general theorem. (Contributed by ML, 17-Jul-2020)

	Ref	Expression
Assertion	exellim	$$⊢\left(\exists xx\in A\wedge \forall x\left(x\in A\to \phi \right)\right)\to \phi$$

Proof

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