Theorem bj-19.36im

Description: One direction of 19.36 from the same axioms as 19.36imv . (Contributed by BJ, 2-Dec-2023)

	Ref	Expression
Assertion	bj-19.36im	$$⊢Ⅎ\text{'}x\psi \to \left(\exists x\left(\phi \to \psi \right)\to \left(\forall x\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		19.35	$$⊢\exists x\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\phi \to \exists x\psi \right)$$
2		bj-nnfe	$$⊢Ⅎ\text{'}x\psi \to \left(\exists x\psi \to \psi \right)$$
3	2	imim2d	$$⊢Ⅎ\text{'}x\psi \to \left(\left(\forall x\phi \to \exists x\psi \right)\to \left(\forall x\phi \to \psi \right)\right)$$
4	1 3	biimtrid	$$⊢Ⅎ\text{'}x\psi \to \left(\exists x\left(\phi \to \psi \right)\to \left(\forall x\phi \to \psi \right)\right)$$