Theorem negrebd

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

	Ref	Expression
Hypotheses	negidd.1	$$⊢\phi \to A\in ℂ$$
	negrebd.2	$$⊢\phi \to -A\in ℝ$$
Assertion	negrebd	$$⊢\phi \to A\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		negrebd.2	$$⊢\phi \to -A\in ℝ$$
3		negreb	$$⊢A\in ℂ\to \left(-A\in ℝ\leftrightarrow A\in ℝ\right)$$
4	1 3	syl	$$⊢\phi \to \left(-A\in ℝ\leftrightarrow A\in ℝ\right)$$
5	2 4	mpbid	$$⊢\phi \to A\in ℝ$$