Theorem 19.36imv

Description: One direction of 19.36v that can be proven without ax-6 . (Contributed by Rohan Ridenour, 16-Apr-2022) (Proof shortened by Wolf Lammen, 22-Sep-2024)

	Ref	Expression
Assertion	19.36imv	$$⊢\exists x\left(\phi \to \psi \right)\to \left(\forall x\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm2.27	$$⊢\phi \to \left(\left(\phi \to \psi \right)\to \psi \right)$$
2	1	aleximi	$$⊢\forall x\phi \to \left(\exists x\left(\phi \to \psi \right)\to \exists x\psi \right)$$
3		ax5e	$$⊢\exists x\psi \to \psi$$
4	2 3	syl6com	$$⊢\exists x\left(\phi \to \psi \right)\to \left(\forall x\phi \to \psi \right)$$