Theorem bj-nfalt

Description: Closed form of nfal . (Contributed by BJ, 2-May-2019)

	Ref	Expression
Assertion	bj-nfalt	$$⊢\forall xℲy\phi \to Ⅎy\forall x\phi$$

Proof

Step	Hyp	Ref	Expression
1		bj-hbalt	$$⊢\forall x\left(\phi \to \forall y\phi \right)\to \left(\forall x\phi \to \forall y\forall x\phi \right)$$
2	1	alimi	$$⊢\forall y\forall x\left(\phi \to \forall y\phi \right)\to \forall y\left(\forall x\phi \to \forall y\forall x\phi \right)$$
3	2	alcoms	$$⊢\forall x\forall y\left(\phi \to \forall y\phi \right)\to \forall y\left(\forall x\phi \to \forall y\forall x\phi \right)$$
4		nf5	$$⊢Ⅎy\phi \leftrightarrow \forall y\left(\phi \to \forall y\phi \right)$$
5	4	albii	$$⊢\forall xℲy\phi \leftrightarrow \forall x\forall y\left(\phi \to \forall y\phi \right)$$
6		nf5	$$⊢Ⅎy\forall x\phi \leftrightarrow \forall y\left(\forall x\phi \to \forall y\forall x\phi \right)$$
7	3 5 6	3imtr4i	$$⊢\forall xℲy\phi \to Ⅎy\forall x\phi$$