Theorem neanior

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

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

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	anbi12i	$$⊢\left(A\ne B\wedge C\ne D\right)\leftrightarrow \left(\neg A=B\wedge \neg C=D\right)$$
4		pm4.56	$$⊢\left(\neg A=B\wedge \neg C=D\right)\leftrightarrow \neg \left(A=B\vee C=D\right)$$
5	3 4	bitri	$$⊢\left(A\ne B\wedge C\ne D\right)\leftrightarrow \neg \left(A=B\vee C=D\right)$$