Theorem imnang

Description: Quantified implication in terms of quantified negation of conjunction. (Contributed by BJ, 16-Jul-2021)

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

Proof

Step	Hyp	Ref	Expression
1		imnan	$$⊢\left(\phi \to \neg \psi \right)\leftrightarrow \neg \left(\phi \wedge \psi \right)$$
2	1	albii	$$⊢\forall x\left(\phi \to \neg \psi \right)\leftrightarrow \forall x\neg \left(\phi \wedge \psi \right)$$