Theorem reximdva0

Description: Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012)

	Ref	Expression
Hypothesis	reximdva0.1	$$⊢\left(\phi \wedge x\in A\right)\to \psi$$
Assertion	reximdva0	$$⊢\left(\phi \wedge A\ne \varnothing \right)\to \exists x\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		reximdva0.1	$$⊢\left(\phi \wedge x\in A\right)\to \psi$$
2		n0	$$⊢A\ne \varnothing \leftrightarrow \exists xx\in A$$
3	1	ex	$$⊢\phi \to \left(x\in A\to \psi \right)$$
4	3	ancld	$$⊢\phi \to \left(x\in A\to \left(x\in A\wedge \psi \right)\right)$$
5	4	eximdv	$$⊢\phi \to \left(\exists xx\in A\to \exists x\left(x\in A\wedge \psi \right)\right)$$
6	5	imp	$$⊢\left(\phi \wedge \exists xx\in A\right)\to \exists x\left(x\in A\wedge \psi \right)$$
7	2 6	sylan2b	$$⊢\left(\phi \wedge A\ne \varnothing \right)\to \exists x\left(x\in A\wedge \psi \right)$$
8		df-rex	$$⊢\exists x\in A\psi \leftrightarrow \exists x\left(x\in A\wedge \psi \right)$$
9	7 8	sylibr	$$⊢\left(\phi \wedge A\ne \varnothing \right)\to \exists x\in A\psi$$