Theorem r19.36vf

Description: Restricted quantifier version of one direction of 19.36 . (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypothesis	r19.36vf.1	$$⊢Ⅎx\psi$$
Assertion	r19.36vf	$$⊢\exists x\in A\left(\phi \to \psi \right)\to \left(\forall x\in A\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		r19.36vf.1	$$⊢Ⅎx\psi$$
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		idd	$$⊢x\in A\to \left(\psi \to \psi \right)$$
4	1 3	rexlimi	$$⊢\exists x\in A\psi \to \psi$$
5	4	imim2i	$$⊢\left(\forall x\in A\phi \to \exists x\in A\psi \right)\to \left(\forall x\in A\phi \to \psi \right)$$
6	2 5	sylbi	$$⊢\exists x\in A\left(\phi \to \psi \right)\to \left(\forall x\in A\phi \to \psi \right)$$