Theorem subcani

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

	Ref	Expression
Hypotheses	negidi.1	|- A e. CC
	pncan3i.2	|- B e. CC
	subadd.3	|- C e. CC
Assertion	subcani	|- ( ( A - B ) = ( A - C ) <-> B = C )

Proof

Step	Hyp	Ref	Expression
1		negidi.1	|- A e. CC
2		pncan3i.2	|- B e. CC
3		subadd.3	|- C e. CC
4		subcan	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = ( A - C ) <-> B = C ) )
5	1 2 3 4	mp3an	|- ( ( A - B ) = ( A - C ) <-> B = C )