Theorem lesub2d

Description: Subtraction 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 )
	ltadd1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
Assertion	lesub2d	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 − 𝐵 ) ≤ ( 𝐶 − 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		lesub2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 − 𝐵 ) ≤ ( 𝐶 − 𝐴 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 − 𝐵 ) ≤ ( 𝐶 − 𝐴 ) ) )