Theorem absltd

Description: Absolute value and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	absltd.1	$$⊢\phi \to A\in ℝ$$
	absltd.2	$$⊢\phi \to B\in ℝ$$
Assertion	absltd	$$⊢\phi \to \left(\left|A\right|<B\leftrightarrow \left(-B<A\wedge A<B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		absltd.1	$$⊢\phi \to A\in ℝ$$
2		absltd.2	$$⊢\phi \to B\in ℝ$$
3		abslt	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(\left|A\right|<B\leftrightarrow \left(-B<A\wedge A<B\right)\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(\left|A\right|<B\leftrightarrow \left(-B<A\wedge A<B\right)\right)$$