Theorem lenegcon2d

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 )
	lenegcon2d.3	⊢ ( 𝜑 → 𝐴 ≤ - 𝐵 )
Assertion	lenegcon2d	⊢ ( 𝜑 → 𝐵 ≤ - 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		lenegcon2d.3	⊢ ( 𝜑 → 𝐴 ≤ - 𝐵 )
4		lenegcon2	⊢ ( ( 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 )