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Metamath Proof Explorer

Theorem absled

Description: Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses absltd.1 ⊢ φ → A ∈ ℝ
absltd.2 ⊢ φ → B ∈ ℝ
Assertion absled ⊢ φ → A ≤ B ↔ − B ≤ A ∧ A ≤ B

Proof

Step Hyp Ref Expression
1 absltd.1 ⊢ φ → A ∈ ℝ
2 absltd.2 ⊢ φ → B ∈ ℝ
3 absle ⊢ A ∈ ℝ ∧ B ∈ ℝ → A ≤ B ↔ − B ≤ A ∧ A ≤ B
4 1 2 3 syl2anc ⊢ φ → A ≤ B ↔ − B ≤ A ∧ A ≤ B