Theorem norcom

Description: The connector -\/ is commutative. (Contributed by Remi, 25-Oct-2023) (Proof shortened by Wolf Lammen, 23-Apr-2024)

	Ref	Expression
Assertion	norcom	$$⊢\left(\phi ⊽\psi \right)\leftrightarrow \left(\psi ⊽\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-nor	$$⊢\left(\phi ⊽\psi \right)\leftrightarrow \neg \left(\phi \vee \psi \right)$$
2		orcom	$$⊢\left(\phi \vee \psi \right)\leftrightarrow \left(\psi \vee \phi \right)$$
3	1 2	xchbinx	$$⊢\left(\phi ⊽\psi \right)\leftrightarrow \neg \left(\psi \vee \phi \right)$$
4		df-nor	$$⊢\left(\psi ⊽\phi \right)\leftrightarrow \neg \left(\psi \vee \phi \right)$$
5	3 4	bitr4i	$$⊢\left(\phi ⊽\psi \right)\leftrightarrow \left(\psi ⊽\phi \right)$$