Theorem negcon1

Description: Negative contraposition law. (Contributed by NM, 9-May-2004)

	Ref	Expression
Assertion	negcon1	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(-A=B\leftrightarrow -B=A\right)$$

Proof

Step	Hyp	Ref	Expression
1		negcl	$$⊢A\in ℂ\to -A\in ℂ$$
2		neg11	$$⊢\left(-A\in ℂ\wedge B\in ℂ\right)\to \left(-\left(-A\right)=-B\leftrightarrow -A=B\right)$$
3	1 2	sylan	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(-\left(-A\right)=-B\leftrightarrow -A=B\right)$$
4		negneg	$$⊢A\in ℂ\to -\left(-A\right)=A$$
5	4	adantr	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to -\left(-A\right)=A$$
6	5	eqeq1d	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(-\left(-A\right)=-B\leftrightarrow A=-B\right)$$
7	3 6	bitr3d	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(-A=B\leftrightarrow A=-B\right)$$
8		eqcom	$$⊢A=-B\leftrightarrow -B=A$$
9	7 8	bitrdi	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(-A=B\leftrightarrow -B=A\right)$$