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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