Theorem neg11ad

Description: The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 . Generalization of neg11d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	negidd.1	$$⊢\phi \to A\in ℂ$$
	neg11ad.2	$$⊢\phi \to B\in ℂ$$
Assertion	neg11ad	$$⊢\phi \to \left(-A=-B\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		neg11ad.2	$$⊢\phi \to B\in ℂ$$
3		neg11	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(-A=-B\leftrightarrow A=B\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(-A=-B\leftrightarrow A=B\right)$$