Theorem lenegcon2d

Description: Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	$$⊢\phi \to A\in ℝ$$
	ltnegd.2	$$⊢\phi \to B\in ℝ$$
	lenegcon2d.3	$$⊢\phi \to A\le -B$$
Assertion	lenegcon2d	$$⊢\phi \to B\le -A$$

Proof

Step	Hyp	Ref	Expression
1		leidd.1	$$⊢\phi \to A\in ℝ$$
2		ltnegd.2	$$⊢\phi \to B\in ℝ$$
3		lenegcon2d.3	$$⊢\phi \to A\le -B$$
4		lenegcon2	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A\le -B\leftrightarrow B\le -A\right)$$
5	1 2 4	syl2anc	$$⊢\phi \to \left(A\le -B\leftrightarrow B\le -A\right)$$
6	3 5	mpbid	$$⊢\phi \to B\le -A$$