Theorem r19.36v

Description: Restricted quantifier version of one direction of 19.36 . (The other direction holds iff A is nonempty, see r19.36zv .) (Contributed by NM, 22-Oct-2003)

	Ref	Expression
Assertion	r19.36v	$$⊢\exists x\in A\left(\phi \to \psi \right)\to \left(\forall x\in A\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		r19.35	$$⊢\exists x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\phi \to \exists x\in A\psi \right)$$
2		id	$$⊢\psi \to \psi$$
3	2	rexlimivw	$$⊢\exists x\in A\psi \to \psi$$
4	3	imim2i	$$⊢\left(\forall x\in A\phi \to \exists x\in A\psi \right)\to \left(\forall x\in A\phi \to \psi \right)$$
5	1 4	sylbi	$$⊢\exists x\in A\left(\phi \to \psi \right)\to \left(\forall x\in A\phi \to \psi \right)$$