Theorem xorneg1

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	xorneg1	$$⊢\left(\neg \phi ⊻\psi \right)\leftrightarrow \neg \left(\phi ⊻\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		xorcom	$$⊢\left(\neg \phi ⊻\psi \right)\leftrightarrow \left(\psi ⊻\neg \phi \right)$$
2		xorneg2	$$⊢\left(\psi ⊻\neg \phi \right)\leftrightarrow \neg \left(\psi ⊻\phi \right)$$
3		xorcom	$$⊢\left(\psi ⊻\phi \right)\leftrightarrow \left(\phi ⊻\psi \right)$$
4	2 3	xchbinx	$$⊢\left(\psi ⊻\neg \phi \right)\leftrightarrow \neg \left(\phi ⊻\psi \right)$$
5	1 4	bitri	$$⊢\left(\neg \phi ⊻\psi \right)\leftrightarrow \neg \left(\phi ⊻\psi \right)$$