Theorem neg11

Description: Negative is one-to-one. (Contributed by NM, 8-Feb-2005) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	neg11	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(-A=-B\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		negeq	$$⊢-A=-B\to -\left(-A\right)=-\left(-B\right)$$
2		negneg	$$⊢A\in ℂ\to -\left(-A\right)=A$$
3		negneg	$$⊢B\in ℂ\to -\left(-B\right)=B$$
4	2 3	eqeqan12d	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(-\left(-A\right)=-\left(-B\right)\leftrightarrow A=B\right)$$
5	1 4	syl5ib	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(-A=-B\to A=B\right)$$
6		negeq	$$⊢A=B\to -A=-B$$
7	5 6	impbid1	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(-A=-B\leftrightarrow A=B\right)$$