Theorem addcan2i

Description: Cancellation law for addition. Theorem I.1 of Apostol p. 18. (Contributed by NM, 14-May-2003) (Revised by Scott Fenton, 3-Jan-2013)

	Ref	Expression
Hypotheses	mul.1	⊢ 𝐴 ∈ ℂ
	mul.2	⊢ 𝐵 ∈ ℂ
	mul.3	⊢ 𝐶 ∈ ℂ
Assertion	addcan2i	⊢ ( ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) ↔ 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		mul.1	⊢ 𝐴 ∈ ℂ
2		mul.2	⊢ 𝐵 ∈ ℂ
3		mul.3	⊢ 𝐶 ∈ ℂ
4		addcan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) ↔ 𝐴 = 𝐵 ) )
5	1 2 3 4	mp3an	⊢ ( ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) ↔ 𝐴 = 𝐵 )