Metamath Proof Explorer


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 = - 𝐵𝐴 = 𝐵 )

Proof

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