Theorem hbalg

Description: Closed form of hbal . Derived from hbalgVD . (Contributed by Alan Sare, 8-Feb-2014) (Proof modification is discouraged.) (New usage is discouraged.)

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

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	$$⊢\forall y\left(\phi \to \forall x\phi \right)\to \left(\forall y\phi \to \forall x\forall y\phi \right)$$
4	3	axc4i	$$⊢\forall y\left(\phi \to \forall x\phi \right)\to \forall y\left(\forall y\phi \to \forall x\forall y\phi \right)$$