Theorem addsub4i

Description: Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999)

	Ref	Expression
Hypotheses	negidi.1	|- A e. CC
	pncan3i.2	|- B e. CC
	subadd.3	|- C e. CC
	addsub4i.4	|- D e. CC
Assertion	addsub4i	|- ( ( A + B ) - ( C + D ) ) = ( ( A - C ) + ( B - D ) )

Proof

Step	Hyp	Ref	Expression
1		negidi.1	|- A e. CC
2		pncan3i.2	|- B e. CC
3		subadd.3	|- C e. CC
4		addsub4i.4	|- D e. CC
5		addsub4	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) - ( C + D ) ) = ( ( A - C ) + ( B - D ) ) )
6	1 2 3 4 5	mp4an	|- ( ( A + B ) - ( C + D ) ) = ( ( A - C ) + ( B - D ) )