Theorem negned

Description: If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	negidd.1	$$⊢\phi \to A\in ℂ$$
	negned.2	$$⊢\phi \to B\in ℂ$$
	negned.3	$$⊢\phi \to A\ne B$$
Assertion	negned	$$⊢\phi \to -A\ne -B$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		negned.2	$$⊢\phi \to B\in ℂ$$
3		negned.3	$$⊢\phi \to A\ne B$$
4	1 2	neg11ad	$$⊢\phi \to \left(-A=-B\leftrightarrow A=B\right)$$
5	4	necon3bid	$$⊢\phi \to \left(-A\ne -B\leftrightarrow A\ne B\right)$$
6	3 5	mpbird	$$⊢\phi \to -A\ne -B$$