Theorem subcani

Description: Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005)

	Ref	Expression
Hypotheses	negidi.1	$$⊢A\in ℂ$$
	pncan3i.2	$$⊢B\in ℂ$$
	subadd.3	$$⊢C\in ℂ$$
Assertion	subcani	$$⊢A-B=A-C\leftrightarrow B=C$$

Proof

Step	Hyp	Ref	Expression
1		negidi.1	$$⊢A\in ℂ$$
2		pncan3i.2	$$⊢B\in ℂ$$
3		subadd.3	$$⊢C\in ℂ$$
4		subcan	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge C\in ℂ\right)\to \left(A-B=A-C\leftrightarrow B=C\right)$$
5	1 2 3 4	mp3an	$$⊢A-B=A-C\leftrightarrow B=C$$