Theorem suble

Description: Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005)

	Ref	Expression
Assertion	suble	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) ≤ 𝐶 ↔ ( 𝐴 − 𝐶 ) ≤ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		lesubadd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) ≤ 𝐶 ↔ 𝐴 ≤ ( 𝐶 + 𝐵 ) ) )
2		lesubadd2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 − 𝐶 ) ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐶 + 𝐵 ) ) )
3	2	3com23	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐶 ) ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐶 + 𝐵 ) ) )
4	1 3	bitr4d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) ≤ 𝐶 ↔ ( 𝐴 − 𝐶 ) ≤ 𝐵 ) )