Metamath Proof Explorer


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 ( 𝜑𝐴 ∈ ℝ )
ltnegd.2 ( 𝜑𝐵 ∈ ℝ )
Assertion lenegd ( 𝜑 → ( 𝐴𝐵 ↔ - 𝐵 ≤ - 𝐴 ) )

Proof

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