Theorem nannan

Description: Nested alternative denials. (Contributed by Jeff Hoffman, 19-Nov-2007) (Proof shortened by Wolf Lammen, 26-Jun-2020)

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

Proof

Step	Hyp	Ref	Expression
1		dfnan2	$$⊢\left(\phi ⊼\left(\psi ⊼\chi \right)\right)\leftrightarrow \left(\phi \to \neg \left(\psi ⊼\chi \right)\right)$$
2		nanan	$$⊢\left(\psi \wedge \chi \right)\leftrightarrow \neg \left(\psi ⊼\chi \right)$$
3	2	imbi2i	$$⊢\left(\phi \to \left(\psi \wedge \chi \right)\right)\leftrightarrow \left(\phi \to \neg \left(\psi ⊼\chi \right)\right)$$
4	1 3	bitr4i	$$⊢\left(\phi ⊼\left(\psi ⊼\chi \right)\right)\leftrightarrow \left(\phi \to \left(\psi \wedge \chi \right)\right)$$