Theorem negned

Description: If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	negned.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	negned.3	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
Assertion	negned	⊢ ( 𝜑 → - 𝐴 ≠ - 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		negned.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		negned.3	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
4	1 2	neg11ad	⊢ ( 𝜑 → ( - 𝐴 = - 𝐵 ↔ 𝐴 = 𝐵 ) )
5	4	necon3bid	⊢ ( 𝜑 → ( - 𝐴 ≠ - 𝐵 ↔ 𝐴 ≠ 𝐵 ) )
6	3 5	mpbird	⊢ ( 𝜑 → - 𝐴 ≠ - 𝐵 )