Theorem 19.37imv

Description: One direction of 19.37v that can be proven without ax-6 . (Contributed by Rohan Ridenour, 16-Apr-2022)

	Ref	Expression
Assertion	19.37imv	$$⊢\exists x\left(\phi \to \psi \right)\to \left(\phi \to \exists x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-5	$$⊢\phi \to \forall x\phi$$
2		19.35	$$⊢\exists x\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\phi \to \exists x\psi \right)$$
3	2	biimpi	$$⊢\exists x\left(\phi \to \psi \right)\to \left(\forall x\phi \to \exists x\psi \right)$$
4	1 3	syl5	$$⊢\exists x\left(\phi \to \psi \right)\to \left(\phi \to \exists x\psi \right)$$