Theorem bj-hbalt

Description: Closed form of hbal . When in main part, prove hbal and hbald from it. (Contributed by BJ, 2-May-2019)

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

Proof

Step	Hyp	Ref	Expression
1		alim	$$⊢\forall y\left(\phi \to \forall x\phi \right)\to \left(\forall y\phi \to \forall y\forall x\phi \right)$$
2		ax-11	$$⊢\forall y\forall x\phi \to \forall x\forall y\phi$$
3	1 2	syl6	$$⊢\forall y\left(\phi \to \forall x\phi \right)\to \left(\forall y\phi \to \forall x\forall y\phi \right)$$