Theorem ltadd2d

Description: Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	$$⊢\phi \to A\in ℝ$$
	ltd.2	$$⊢\phi \to B\in ℝ$$
	letrd.3	$$⊢\phi \to C\in ℝ$$
Assertion	ltadd2d	$$⊢\phi \to \left(A<B\leftrightarrow C+A<C+B\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltd.1	$$⊢\phi \to A\in ℝ$$
2		ltd.2	$$⊢\phi \to B\in ℝ$$
3		letrd.3	$$⊢\phi \to C\in ℝ$$
4		ltadd2	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge C\in ℝ\right)\to \left(A<B\leftrightarrow C+A<C+B\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(A<B\leftrightarrow C+A<C+B\right)$$