Theorem r19.35

Description: Restricted quantifier version of 19.35 . (Contributed by NM, 20-Sep-2003) (Proof shortened by Wolf Lammen, 22-Dec-2024)

	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		pm5.5	$$⊢\phi \to \left(\left(\phi \to \psi \right)\leftrightarrow \psi \right)$$
2	1	ralrexbid	$$⊢\forall x\in A\phi \to \left(\exists x\in A\left(\phi \to \psi \right)\leftrightarrow \exists x\in A\psi \right)$$
3	2	biimpcd	$$⊢\exists x\in A\left(\phi \to \psi \right)\to \left(\forall x\in A\phi \to \exists x\in A\psi \right)$$
4		rexnal	$$⊢\exists x\in A\neg \phi \leftrightarrow \neg \forall x\in A\phi$$
5		pm2.21	$$⊢\neg \phi \to \left(\phi \to \psi \right)$$
6	5	reximi	$$⊢\exists x\in A\neg \phi \to \exists x\in A\left(\phi \to \psi \right)$$
7	4 6	sylbir	$$⊢\neg \forall x\in A\phi \to \exists x\in A\left(\phi \to \psi \right)$$
8		ax-1	$$⊢\psi \to \left(\phi \to \psi \right)$$
9	8	reximi	$$⊢\exists x\in A\psi \to \exists x\in A\left(\phi \to \psi \right)$$
10	7 9	ja	$$⊢\left(\forall x\in A\phi \to \exists x\in A\psi \right)\to \exists x\in A\left(\phi \to \psi \right)$$
11	3 10	impbii	$$⊢\exists x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\phi \to \exists x\in A\psi \right)$$