Theorem ltnegd

Description: Negative of both sides of 'less than'. Theorem I.23 of Apostol p. 20. (Contributed by Mario Carneiro, 27-May-2016)

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

Proof

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