Theorem xor2

Description: Two ways to express "exclusive or". (Contributed by Mario Carneiro, 4-Sep-2016)

	Ref	Expression
Assertion	xor2	$$⊢\left(\phi ⊻\psi \right)\leftrightarrow \left(\left(\phi \vee \psi \right)\wedge \neg \left(\phi \wedge \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-xor	$$⊢\left(\phi ⊻\psi \right)\leftrightarrow \neg \left(\phi \leftrightarrow \psi \right)$$
2		nbi2	$$⊢\neg \left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\left(\phi \vee \psi \right)\wedge \neg \left(\phi \wedge \psi \right)\right)$$
3	1 2	bitri	$$⊢\left(\phi ⊻\psi \right)\leftrightarrow \left(\left(\phi \vee \psi \right)\wedge \neg \left(\phi \wedge \psi \right)\right)$$