Theorem xorneg2

Description: The connector \/_ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 27-Jun-2020)

	Ref	Expression
Assertion	xorneg2	$$⊢\left(\phi ⊻\neg \psi \right)\leftrightarrow \neg \left(\phi ⊻\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-xor	$$⊢\left(\phi ⊻\neg \psi \right)\leftrightarrow \neg \left(\phi \leftrightarrow \neg \psi \right)$$
2		pm5.18	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \neg \left(\phi \leftrightarrow \neg \psi \right)$$
3		xnor	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \neg \left(\phi ⊻\psi \right)$$
4	1 2 3	3bitr2i	$$⊢\left(\phi ⊻\neg \psi \right)\leftrightarrow \neg \left(\phi ⊻\psi \right)$$