Theorem lenegd

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

	Ref	Expression
Hypotheses	leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	lenegd	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ - 𝐵 ≤ - 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		leneg	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ - 𝐵 ≤ - 𝐴 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ - 𝐵 ≤ - 𝐴 ) )