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	$$⊢A\in ℂ$$
	mul.2	$$⊢B\in ℂ$$
	mul.3	$$⊢C\in ℂ$$
Assertion	addcani	$$⊢A+B=A+C\leftrightarrow B=C$$

Proof

Step	Hyp	Ref	Expression
1		mul.1	$$⊢A\in ℂ$$
2		mul.2	$$⊢B\in ℂ$$
3		mul.3	$$⊢C\in ℂ$$
4		addcan	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge C\in ℂ\right)\to \left(A+B=A+C\leftrightarrow B=C\right)$$
5	1 2 3 4	mp3an	$$⊢A+B=A+C\leftrightarrow B=C$$