Theorem nbi2

Description: Two ways to express "exclusive or". (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 24-Jan-2013)

	Ref	Expression
Assertion	nbi2	$$⊢\neg \left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\left(\phi \vee \psi \right)\wedge \neg \left(\phi \wedge \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		xor3	$$⊢\neg \left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\phi \leftrightarrow \neg \psi \right)$$
2		pm5.17	$$⊢\left(\left(\phi \vee \psi \right)\wedge \neg \left(\phi \wedge \psi \right)\right)\leftrightarrow \left(\phi \leftrightarrow \neg \psi \right)$$
3	1 2	bitr4i	$$⊢\neg \left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\left(\phi \vee \psi \right)\wedge \neg \left(\phi \wedge \psi \right)\right)$$