Theorem neg11i

Description: Negative is one-to-one. (Contributed by NM, 1-Aug-1999)

	Ref	Expression
Hypotheses	negidi.1	⊢ 𝐴 ∈ ℂ
	pncan3i.2	⊢ 𝐵 ∈ ℂ
Assertion	neg11i	⊢ ( - 𝐴 = - 𝐵 ↔ 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		negidi.1	⊢ 𝐴 ∈ ℂ
2		pncan3i.2	⊢ 𝐵 ∈ ℂ
3		neg11	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 = - 𝐵 ↔ 𝐴 = 𝐵 ) )
4	1 2 3	mp2an	⊢ ( - 𝐴 = - 𝐵 ↔ 𝐴 = 𝐵 )