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 )