Theorem exellimddv

Description: Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim for the closed form, which requires the use of a universal quantifier. (Contributed by ML, 17-Jul-2020)

	Ref	Expression
Hypotheses	exellimddv.1	$$⊢\phi \to \exists xx\in A$$
	exellimddv.2	$$⊢\phi \to \left(x\in A\to \psi \right)$$
Assertion	exellimddv	$$⊢\phi \to \psi$$

Proof

Step	Hyp	Ref	Expression
1		exellimddv.1	$$⊢\phi \to \exists xx\in A$$
2		exellimddv.2	$$⊢\phi \to \left(x\in A\to \psi \right)$$
3	2	alrimiv	$$⊢\phi \to \forall x\left(x\in A\to \psi \right)$$
4		exellim	$$⊢\left(\exists xx\in A\wedge \forall x\left(x\in A\to \psi \right)\right)\to \psi$$
5	1 3 4	syl2anc	$$⊢\phi \to \psi$$