Theorem rexlimi

Description: Restricted quantifier version of exlimi . (Contributed by NM, 30-Nov-2003) (Proof shortened by Andrew Salmon, 30-May-2011)

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

Proof

Step	Hyp	Ref	Expression
1		rexlimi.1	$$⊢Ⅎx\psi$$
2		rexlimi.2	$$⊢x\in A\to \left(\phi \to \psi \right)$$
3	2	rgen	$$⊢\forall x\in A\left(\phi \to \psi \right)$$
4	1	r19.23	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\in A\phi \to \psi \right)$$
5	3 4	mpbi	$$⊢\exists x\in A\phi \to \psi$$