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Metamath Proof Explorer

Theorem lenegcon1d

Description: Contraposition of negative in '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
lenegcon1d.3 ⊢ φ → − A ≤ B
Assertion lenegcon1d ⊢ φ → − B ≤ A

Proof

Step Hyp Ref Expression
1 leidd.1 ⊢ φ → A ∈ ℝ
2 ltnegd.2 ⊢ φ → B ∈ ℝ
3 lenegcon1d.3 ⊢ φ → − A ≤ B
4 lenegcon1 ⊢ A ∈ ℝ ∧ B ∈ ℝ → − A ≤ B ↔ − B ≤ A
5 1 2 4 syl2anc ⊢ φ → − A ≤ B ↔ − B ≤ A
6 3 5 mpbid ⊢ φ → − B ≤ A