Theorem bj-hbext

Description: Closed form of bj-hbex and hbex . (Contributed by BJ, 10-Oct-2019)

	Ref	Expression
Assertion	bj-hbext	$$⊢\forall y\forall x\left(\phi \to \forall x\psi \right)\to \left(\exists y\phi \to \forall x\exists y\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢\forall y\forall x\left(\phi \to \forall x\psi \right)\to \forall y\forall x\left(\phi \to \forall x\psi \right)$$
2		sp	$$⊢\forall x\left(\phi \to \forall x\psi \right)\to \left(\phi \to \forall x\psi \right)$$
3	1 2	bj-hbexd	$$⊢\forall y\forall x\left(\phi \to \forall x\psi \right)\to \left(\exists y\phi \to \forall x\exists y\psi \right)$$