Theorem subcand

Description: Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	negidd.1	$$⊢\phi \to A\in ℂ$$
	pncand.2	$$⊢\phi \to B\in ℂ$$
	subaddd.3	$$⊢\phi \to C\in ℂ$$
	subcand.4	$$⊢\phi \to A-B=A-C$$
Assertion	subcand	$$⊢\phi \to B=C$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		pncand.2	$$⊢\phi \to B\in ℂ$$
3		subaddd.3	$$⊢\phi \to C\in ℂ$$
4		subcand.4	$$⊢\phi \to A-B=A-C$$
5		subcan	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge C\in ℂ\right)\to \left(A-B=A-C\leftrightarrow B=C\right)$$
6	1 2 3 5	syl3anc	$$⊢\phi \to \left(A-B=A-C\leftrightarrow B=C\right)$$
7	4 6	mpbid	$$⊢\phi \to B=C$$