Theorem subaddi

Description: Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994) (Revised by Mario Carneiro, 21-Dec-2013)

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

Proof

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