Theorem xorcom

Description: The connector \/_ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 21-Apr-2024)

	Ref	Expression
Assertion	xorcom	$$⊢\left(\phi ⊻\psi \right)\leftrightarrow \left(\psi ⊻\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-xor	$$⊢\left(\phi ⊻\psi \right)\leftrightarrow \neg \left(\phi \leftrightarrow \psi \right)$$
2		bicom	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\psi \leftrightarrow \phi \right)$$
3	1 2	xchbinx	$$⊢\left(\phi ⊻\psi \right)\leftrightarrow \neg \left(\psi \leftrightarrow \phi \right)$$
4		df-xor	$$⊢\left(\psi ⊻\phi \right)\leftrightarrow \neg \left(\psi \leftrightarrow \phi \right)$$
5	3 4	bitr4i	$$⊢\left(\phi ⊻\psi \right)\leftrightarrow \left(\psi ⊻\phi \right)$$