Theorem xfree2

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

	Ref	Expression
Assertion	xfree2	$$⊢\forall x\left(\phi \to \forall x\phi \right)\leftrightarrow \forall x\left(\neg \phi \to \forall x\neg \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		xfree	$$⊢\forall x\left(\phi \to \forall x\phi \right)\leftrightarrow \forall x\left(\exists x\phi \to \phi \right)$$
2		eximal	$$⊢\left(\exists x\phi \to \phi \right)\leftrightarrow \left(\neg \phi \to \forall x\neg \phi \right)$$
3	2	albii	$$⊢\forall x\left(\exists x\phi \to \phi \right)\leftrightarrow \forall x\left(\neg \phi \to \forall x\neg \phi \right)$$
4	1 3	bitri	$$⊢\forall x\left(\phi \to \forall x\phi \right)\leftrightarrow \forall x\left(\neg \phi \to \forall x\neg \phi \right)$$