Theorem bj-hbntbi

Description: Strengthening hbnt by replacing its succedent with a biconditional. See also hbntg and hbntal . (Contributed by BJ, 20-Oct-2019) Proved from bj-19.9htbi . (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-19.9htbi	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to \left(\exists x\phi \leftrightarrow \phi \right)$$
2	1	bicomd	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to \left(\phi \leftrightarrow \exists x\phi \right)$$
3	2	notbid	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to \left(\neg \phi \leftrightarrow \neg \exists x\phi \right)$$
4		alnex	$$⊢\forall x\neg \phi \leftrightarrow \neg \exists x\phi$$
5	3 4	bitr4di	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to \left(\neg \phi \leftrightarrow \forall x\neg \phi \right)$$