Theorem rerebd

Description: A real number equals its real part. Proposition 10-3.4(f) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	recld.1	$$⊢\phi \to A\in ℂ$$
	rerebd.2	$$⊢\phi \to \Re \left(A\right)=A$$
Assertion	rerebd	$$⊢\phi \to A\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		recld.1	$$⊢\phi \to A\in ℂ$$
2		rerebd.2	$$⊢\phi \to \Re \left(A\right)=A$$
3		rereb	$$⊢A\in ℂ\to \left(A\in ℝ\leftrightarrow \Re \left(A\right)=A\right)$$
4	1 3	syl	$$⊢\phi \to \left(A\in ℝ\leftrightarrow \Re \left(A\right)=A\right)$$
5	2 4	mpbird	$$⊢\phi \to A\in ℝ$$