Metamath Proof Explorer


Theorem negcon1

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

Ref Expression
Assertion negcon1 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 = 𝐵 ↔ - 𝐵 = 𝐴 ) )

Proof

Step Hyp Ref Expression
1 negcl ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
2 neg11 ( ( - 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - - 𝐴 = - 𝐵 ↔ - 𝐴 = 𝐵 ) )
3 1 2 sylan ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - - 𝐴 = - 𝐵 ↔ - 𝐴 = 𝐵 ) )
4 negneg ( 𝐴 ∈ ℂ → - - 𝐴 = 𝐴 )
5 4 adantr ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - - 𝐴 = 𝐴 )
6 5 eqeq1d ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - - 𝐴 = - 𝐵𝐴 = - 𝐵 ) )
7 3 6 bitr3d ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 = 𝐵𝐴 = - 𝐵 ) )
8 eqcom ( 𝐴 = - 𝐵 ↔ - 𝐵 = 𝐴 )
9 7 8 bitrdi ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 = 𝐵 ↔ - 𝐵 = 𝐴 ) )