Theorem addcan2ad

Description: Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	muld.1	$$⊢\phi \to A\in ℂ$$
	addcomd.2	$$⊢\phi \to B\in ℂ$$
	addcand.3	$$⊢\phi \to C\in ℂ$$
	addcan2ad.4	$$⊢\phi \to A+C=B+C$$
Assertion	addcan2ad	$$⊢\phi \to A=B$$

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		addcan2ad.4	$$⊢\phi \to A+C=B+C$$
5	1 2 3	addcan2d	$$⊢\phi \to \left(A+C=B+C\leftrightarrow A=B\right)$$
6	4 5	mpbid	$$⊢\phi \to A=B$$