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 = - 𝐵 ↔ 𝐴 = - 𝐵 ) )
7	3 6	bitr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 = 𝐵 ↔ 𝐴 = - 𝐵 ) )
8		eqcom	⊢ ( 𝐴 = - 𝐵 ↔ - 𝐵 = 𝐴 )
9	7 8	bitrdi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 = 𝐵 ↔ - 𝐵 = 𝐴 ) )