Theorem ltsub23

Description: 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999)

	Ref	Expression
Assertion	ltsub23	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) < 𝐶 ↔ ( 𝐴 − 𝐶 ) < 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ltsubadd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) < 𝐶 ↔ 𝐴 < ( 𝐶 + 𝐵 ) ) )
2		ltsubadd2	⊢ ( ( 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 ) )