Theorem add42i

Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999) (Proof shortened by OpenAI, 25-Mar-2020)

	Ref	Expression
Hypotheses	add.1	|- A e. CC
	add.2	|- B e. CC
	add.3	|- C e. CC
	add4.4	|- D e. CC
Assertion	add42i	|- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) )

Proof

Step	Hyp	Ref	Expression
1		add.1	|- A e. CC
2		add.2	|- B e. CC
3		add.3	|- C e. CC
4		add4.4	|- D e. CC
5		add42	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) )
6	1 2 3 4 5	mp4an	|- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) )