Theorem addcand

Description: Cancellation law for addition. Theorem I.1 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	muld.1	$$⊢\phi \to A\in ℂ$$
	addcomd.2	$$⊢\phi \to B\in ℂ$$
	addcand.3	$$⊢\phi \to C\in ℂ$$
Assertion	addcand	$$⊢\phi \to \left(A+B=A+C\leftrightarrow B=C\right)$$

Proof

Step	Hyp	Ref	Expression
1		muld.1	$$⊢\phi \to A\in ℂ$$
2		addcomd.2	$$⊢\phi \to B\in ℂ$$
3		addcand.3	$$⊢\phi \to C\in ℂ$$
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	syl3anc	$$⊢\phi \to \left(A+B=A+C\leftrightarrow B=C\right)$$