Theorem lesubd

Description: Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	ltadd1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	lesubd.4	⊢ ( 𝜑 → 𝐴 ≤ ( 𝐵 − 𝐶 ) )
Assertion	lesubd	⊢ ( 𝜑 → 𝐶 ≤ ( 𝐵 − 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		ltadd1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		lesubd.4	⊢ ( 𝜑 → 𝐴 ≤ ( 𝐵 − 𝐶 ) )
5		lesub	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ≤ ( 𝐵 − 𝐶 ) ↔ 𝐶 ≤ ( 𝐵 − 𝐴 ) ) )
6	1 2 3 5	syl3anc	⊢ ( 𝜑 → ( 𝐴 ≤ ( 𝐵 − 𝐶 ) ↔ 𝐶 ≤ ( 𝐵 − 𝐴 ) ) )
7	4 6	mpbid	⊢ ( 𝜑 → 𝐶 ≤ ( 𝐵 − 𝐴 ) )