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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 )