Theorem negrebd

Description: The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	negrebd.2	⊢ ( 𝜑 → - 𝐴 ∈ ℝ )
Assertion	negrebd	⊢ ( 𝜑 → 𝐴 ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		negrebd.2	⊢ ( 𝜑 → - 𝐴 ∈ ℝ )
3		negreb	⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ ) )
4	1 3	syl	⊢ ( 𝜑 → ( - 𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ ) )
5	2 4	mpbid	⊢ ( 𝜑 → 𝐴 ∈ ℝ )