Theorem xfree

Description: A partial converse to 19.9t . (Contributed by Stefan Allan, 21-Dec-2008) (Revised by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Assertion	xfree	$$⊢\forall x\left(\phi \to \forall x\phi \right)\leftrightarrow \forall x\left(\exists x\phi \to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nf5	$$⊢Ⅎx\phi \leftrightarrow \forall x\left(\phi \to \forall x\phi \right)$$
2		nf6	$$⊢Ⅎx\phi \leftrightarrow \forall x\left(\exists x\phi \to \phi \right)$$
3	1 2	bitr3i	$$⊢\forall x\left(\phi \to \forall x\phi \right)\leftrightarrow \forall x\left(\exists x\phi \to \phi \right)$$