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Metamath Proof Explorer

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 )