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Metamath Proof Explorer

Theorem eqnegad

Description: If a complex number equals its own negative, it is zero. One-way deduction form of eqneg . (Contributed by David Moews, 28-Feb-2017)

Ref Expression
Hypotheses eqnegad.1 ⊢ φ → A ∈ ℂ
eqnegad.2 ⊢ φ → A = − A
Assertion eqnegad ⊢ φ → A = 0

Proof

Step Hyp Ref Expression
1 eqnegad.1 ⊢ φ → A ∈ ℂ
2 eqnegad.2 ⊢ φ → A = − A
3 1 eqnegd ⊢ φ → A = − A ↔ A = 0
4 2 3 mpbid ⊢ φ → A = 0