Theorem r19.35

Description: Restricted quantifier version of 19.35 . (Contributed by NM, 20-Sep-2003)

	Ref	Expression
Assertion	r19.35	$$⊢\exists x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\phi \to \exists x\in A\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rexim	$$⊢\forall x\in A\left(\left(\phi \to \psi \right)\to \psi \right)\to \left(\exists x\in A\left(\phi \to \psi \right)\to \exists x\in A\psi \right)$$
2		pm2.27	$$⊢\phi \to \left(\left(\phi \to \psi \right)\to \psi \right)$$
3	2	ralimi	$$⊢\forall x\in A\phi \to \forall x\in A\left(\left(\phi \to \psi \right)\to \psi \right)$$
4	1 3	syl11	$$⊢\exists x\in A\left(\phi \to \psi \right)\to \left(\forall x\in A\phi \to \exists x\in A\psi \right)$$
5		rexnal	$$⊢\exists x\in A\neg \phi \leftrightarrow \neg \forall x\in A\phi$$
6		pm2.21	$$⊢\neg \phi \to \left(\phi \to \psi \right)$$
7	6	reximi	$$⊢\exists x\in A\neg \phi \to \exists x\in A\left(\phi \to \psi \right)$$
8	5 7	sylbir	$$⊢\neg \forall x\in A\phi \to \exists x\in A\left(\phi \to \psi \right)$$
9		ax-1	$$⊢\psi \to \left(\phi \to \psi \right)$$
10	9	reximi	$$⊢\exists x\in A\psi \to \exists x\in A\left(\phi \to \psi \right)$$
11	8 10	ja	$$⊢\left(\forall x\in A\phi \to \exists x\in A\psi \right)\to \exists x\in A\left(\phi \to \psi \right)$$
12	4 11	impbii	$$⊢\exists x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\phi \to \exists x\in A\psi \right)$$