Theorem r19.45v

Description: Restricted quantifier version of one direction of 19.45 . The other direction holds when A is nonempty, see r19.45zv . (Contributed by NM, 2-Apr-2004)

	Ref	Expression
Assertion	r19.45v	$$⊢\exists x\in A\left(\phi \vee \psi \right)\to \left(\phi \vee \exists x\in A\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		r19.43	$$⊢\exists x\in A\left(\phi \vee \psi \right)\leftrightarrow \left(\exists x\in A\phi \vee \exists x\in A\psi \right)$$
2		id	$$⊢\phi \to \phi$$
3	2	rexlimivw	$$⊢\exists x\in A\phi \to \phi$$
4	3	orim1i	$$⊢\left(\exists x\in A\phi \vee \exists x\in A\psi \right)\to \left(\phi \vee \exists x\in A\psi \right)$$
5	1 4	sylbi	$$⊢\exists x\in A\left(\phi \vee \psi \right)\to \left(\phi \vee \exists x\in A\psi \right)$$