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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 ⊢ 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