Theorem addcani

Description: Cancellation law for addition. Theorem I.1 of Apostol p. 18. (Contributed by NM, 27-Oct-1999) (Revised by Scott Fenton, 3-Jan-2013)

	Ref	Expression
Hypotheses	mul.1	⊢ 𝐴 ∈ ℂ
	mul.2	⊢ 𝐵 ∈ ℂ
	mul.3	⊢ 𝐶 ∈ ℂ
Assertion	addcani	⊢ ( ( 𝐴 + 𝐵 ) = ( 𝐴 + 𝐶 ) ↔ 𝐵 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		mul.1	⊢ 𝐴 ∈ ℂ
2		mul.2	⊢ 𝐵 ∈ ℂ
3		mul.3	⊢ 𝐶 ∈ ℂ
4		addcan	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) = ( 𝐴 + 𝐶 ) ↔ 𝐵 = 𝐶 ) )
5	1 2 3 4	mp3an	⊢ ( ( 𝐴 + 𝐵 ) = ( 𝐴 + 𝐶 ) ↔ 𝐵 = 𝐶 )