Theorem r19.37

Description: Restricted quantifier version of one direction of 19.37 . (The other direction does not hold when A is empty.) (Contributed by FL, 13-May-2012) (Revised by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Hypothesis	r19.37.1	$$⊢Ⅎx\phi$$
Assertion	r19.37	$$⊢\exists x\in A\left(\phi \to \psi \right)\to \left(\phi \to \exists x\in A\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		r19.37.1	$$⊢Ⅎx\phi$$
2		r19.35	$$⊢\exists x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\phi \to \exists x\in A\psi \right)$$
3		ax-1	$$⊢\phi \to \left(x\in A\to \phi \right)$$
4	1 3	ralrimi	$$⊢\phi \to \forall x\in A\phi$$
5	4	imim1i	$$⊢\left(\forall x\in A\phi \to \exists x\in A\psi \right)\to \left(\phi \to \exists x\in A\psi \right)$$
6	2 5	sylbi	$$⊢\exists x\in A\left(\phi \to \psi \right)\to \left(\phi \to \exists x\in A\psi \right)$$