Theorem nan

Description: Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003)

	Ref	Expression
Assertion	nan	$$⊢\left(\phi \to \neg \left(\psi \wedge \chi \right)\right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\to \neg \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		impexp	$$⊢\left(\left(\phi \wedge \psi \right)\to \neg \chi \right)\leftrightarrow \left(\phi \to \left(\psi \to \neg \chi \right)\right)$$
2		imnan	$$⊢\left(\psi \to \neg \chi \right)\leftrightarrow \neg \left(\psi \wedge \chi \right)$$
3	2	imbi2i	$$⊢\left(\phi \to \left(\psi \to \neg \chi \right)\right)\leftrightarrow \left(\phi \to \neg \left(\psi \wedge \chi \right)\right)$$
4	1 3	bitr2i	$$⊢\left(\phi \to \neg \left(\psi \wedge \chi \right)\right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\to \neg \chi \right)$$