Theorem neorian

Description: A De Morgan's law for inequality. (Contributed by NM, 18-May-2007)

	Ref	Expression
Assertion	neorian	$$⊢\left(A\ne B\vee C\ne D\right)\leftrightarrow \neg \left(A=B\wedge C=D\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-ne	$$⊢A\ne B\leftrightarrow \neg A=B$$
2		df-ne	$$⊢C\ne D\leftrightarrow \neg C=D$$
3	1 2	orbi12i	$$⊢\left(A\ne B\vee C\ne D\right)\leftrightarrow \left(\neg A=B\vee \neg C=D\right)$$
4		ianor	$$⊢\neg \left(A=B\wedge C=D\right)\leftrightarrow \left(\neg A=B\vee \neg C=D\right)$$
5	3 4	bitr4i	$$⊢\left(A\ne B\vee C\ne D\right)\leftrightarrow \neg \left(A=B\wedge C=D\right)$$