Theorem negcon2

Description: Negative contraposition law. (Contributed by NM, 14-Nov-2004)

	Ref	Expression
Assertion	negcon2	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(A=-B\leftrightarrow B=-A\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqcom	$$⊢A=-B\leftrightarrow -B=A$$
2		negcon1	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(-A=B\leftrightarrow -B=A\right)$$
3	1 2	bitr4id	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(A=-B\leftrightarrow -A=B\right)$$
4		eqcom	$$⊢-A=B\leftrightarrow B=-A$$
5	3 4	bitrdi	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(A=-B\leftrightarrow B=-A\right)$$