Theorem ltsub13

Description: 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004)

	Ref	Expression
Assertion	ltsub13	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < ( 𝐵 − 𝐶 ) ↔ 𝐶 < ( 𝐵 − 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltaddsub	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 + 𝐶 ) < 𝐵 ↔ 𝐴 < ( 𝐵 − 𝐶 ) ) )
2		ltaddsub2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 + 𝐶 ) < 𝐵 ↔ 𝐶 < ( 𝐵 − 𝐴 ) ) )
3	1 2	bitr3d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < ( 𝐵 − 𝐶 ) ↔ 𝐶 < ( 𝐵 − 𝐴 ) ) )
4	3	3com23	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < ( 𝐵 − 𝐶 ) ↔ 𝐶 < ( 𝐵 − 𝐴 ) ) )