Theorem subled

Description: Swap subtrahends in an inequality. (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	$$⊢\phi \to C\in ℝ$$
	subled.4	$$⊢\phi \to A-B\le C$$
Assertion	subled	$$⊢\phi \to A-C\le B$$

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		subled.4	$$⊢\phi \to A-B\le C$$
5		suble	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge C\in ℝ\right)\to \left(A-B\le C\leftrightarrow A-C\le B\right)$$
6	1 2 3 5	syl3anc	$$⊢\phi \to \left(A-B\le C\leftrightarrow A-C\le B\right)$$
7	4 6	mpbid	$$⊢\phi \to A-C\le B$$