Theorem lesub1d

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

	Ref	Expression
Hypotheses	leidd.1	$$⊢\phi \to A\in ℝ$$
	ltnegd.2	$$⊢\phi \to B\in ℝ$$
	ltadd1d.3	$$⊢\phi \to C\in ℝ$$
Assertion	lesub1d	$$⊢\phi \to \left(A\le B\leftrightarrow A-C\le B-C\right)$$

Proof

Step	Hyp	Ref	Expression
1		leidd.1	$$⊢\phi \to A\in ℝ$$
2		ltnegd.2	$$⊢\phi \to B\in ℝ$$
3		ltadd1d.3	$$⊢\phi \to C\in ℝ$$
4		lesub1	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge C\in ℝ\right)\to \left(A\le B\leftrightarrow A-C\le B-C\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(A\le B\leftrightarrow A-C\le B-C\right)$$