Theorem negne0bd

Description: A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypothesis	negidd.1	$$⊢\phi \to A\in ℂ$$
Assertion	negne0bd	$$⊢\phi \to \left(A\ne 0\leftrightarrow -A\ne 0\right)$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2	1	negeq0d	$$⊢\phi \to \left(A=0\leftrightarrow -A=0\right)$$
3	2	necon3bid	$$⊢\phi \to \left(A\ne 0\leftrightarrow -A\ne 0\right)$$