Theorem ltnegcon2d

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

	Ref	Expression
Hypotheses	leidd.1	$$⊢\phi \to A\in ℝ$$
	ltnegd.2	$$⊢\phi \to B\in ℝ$$
	ltnegcon2d.3	$$⊢\phi \to A<-B$$
Assertion	ltnegcon2d	$$⊢\phi \to B<-A$$

Proof

Step	Hyp	Ref	Expression
1		leidd.1	$$⊢\phi \to A\in ℝ$$
2		ltnegd.2	$$⊢\phi \to B\in ℝ$$
3		ltnegcon2d.3	$$⊢\phi \to A<-B$$
4		ltnegcon2	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A<-B\leftrightarrow B<-A\right)$$
5	1 2 4	syl2anc	$$⊢\phi \to \left(A<-B\leftrightarrow B<-A\right)$$
6	3 5	mpbid	$$⊢\phi \to B<-A$$