Theorem nancom

Description: Alternative denial is commutative. Remark: alternative denial is not associative, see nanass . (Contributed by Mario Carneiro, 9-May-2015) (Proof shortened by Wolf Lammen, 26-Jun-2020)

	Ref	Expression
Assertion	nancom	$$⊢\left(\phi ⊼\psi \right)\leftrightarrow \left(\psi ⊼\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		con2b	$$⊢\left(\phi \to \neg \psi \right)\leftrightarrow \left(\psi \to \neg \phi \right)$$
2		nanimn	$$⊢\left(\phi ⊼\psi \right)\leftrightarrow \left(\phi \to \neg \psi \right)$$
3		nanimn	$$⊢\left(\psi ⊼\phi \right)\leftrightarrow \left(\psi \to \neg \phi \right)$$
4	1 2 3	3bitr4i	$$⊢\left(\phi ⊼\psi \right)\leftrightarrow \left(\psi ⊼\phi \right)$$