Theorem addcanad

Description: Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand . (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	$$⊢\phi \to C\in ℂ$$
	addcanad.4	$$⊢\phi \to A+B=A+C$$
Assertion	addcanad	$$⊢\phi \to B=C$$

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