Theorem bj-nexdh

Description: Closed form of nexdh (actually, its general instance). (Contributed by BJ, 6-May-2019)

	Ref	Expression
Assertion	bj-nexdh	$$⊢\forall x\left(\phi \to \neg \psi \right)\to \left(\left(\chi \to \forall x\phi \right)\to \left(\chi \to \neg \exists x\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sylgt	$$⊢\forall x\left(\phi \to \neg \psi \right)\to \left(\left(\chi \to \forall x\phi \right)\to \left(\chi \to \forall x\neg \psi \right)\right)$$
2		alnex	$$⊢\forall x\neg \psi \leftrightarrow \neg \exists x\psi$$
3	1 2	syl8ib	$$⊢\forall x\left(\phi \to \neg \psi \right)\to \left(\left(\chi \to \forall x\phi \right)\to \left(\chi \to \neg \exists x\psi \right)\right)$$