Theorem hbexg

Description: Closed form of nfex . Derived from hbexgVD . (Contributed by Alan Sare, 8-Feb-2014) (Revised by Mario Carneiro, 12-Dec-2016) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	hbexg	$$⊢\forall x\forall y\left(\phi \to \forall x\phi \right)\to \forall x\forall y\left(\exists y\phi \to \forall x\exists y\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfa2	$$⊢Ⅎy\forall x\forall y\left(\phi \to \forall x\phi \right)$$
2		sp	$$⊢\forall y\left(\phi \to \forall x\phi \right)\to \left(\phi \to \forall x\phi \right)$$
3	2	alimi	$$⊢\forall x\forall y\left(\phi \to \forall x\phi \right)\to \forall x\left(\phi \to \forall x\phi \right)$$
4		nf5	$$⊢Ⅎx\phi \leftrightarrow \forall x\left(\phi \to \forall x\phi \right)$$
5	3 4	sylibr	$$⊢\forall x\forall y\left(\phi \to \forall x\phi \right)\to Ⅎx\phi$$
6	1 5	nfexd	$$⊢\forall x\forall y\left(\phi \to \forall x\phi \right)\to Ⅎx\exists y\phi$$
7		nf5	$$⊢Ⅎx\exists y\phi \leftrightarrow \forall x\left(\exists y\phi \to \forall x\exists y\phi \right)$$
8	6 7	sylib	$$⊢\forall x\forall y\left(\phi \to \forall x\phi \right)\to \forall x\left(\exists y\phi \to \forall x\exists y\phi \right)$$
9	1 8	alrimi	$$⊢\forall x\forall y\left(\phi \to \forall x\phi \right)\to \forall y\forall x\left(\exists y\phi \to \forall x\exists y\phi \right)$$
10		alcom	$$⊢\forall y\forall x\left(\exists y\phi \to \forall x\exists y\phi \right)\leftrightarrow \forall x\forall y\left(\exists y\phi \to \forall x\exists y\phi \right)$$
11	9 10	sylib	$$⊢\forall x\forall y\left(\phi \to \forall x\phi \right)\to \forall x\forall y\left(\exists y\phi \to \forall x\exists y\phi \right)$$