Theorem negcon2

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

	Ref	Expression
Assertion	negcon2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 = - 𝐵 ↔ 𝐵 = - 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		eqcom	⊢ ( 𝐴 = - 𝐵 ↔ - 𝐵 = 𝐴 )
2		negcon1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 = 𝐵 ↔ - 𝐵 = 𝐴 ) )
3	1 2	bitr4id	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 = - 𝐵 ↔ - 𝐴 = 𝐵 ) )
4		eqcom	⊢ ( - 𝐴 = 𝐵 ↔ 𝐵 = - 𝐴 )
5	3 4	bitrdi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 = - 𝐵 ↔ 𝐵 = - 𝐴 ) )