Theorem add42d

Description: Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	addd.1	|- ( ph -> A e. CC )
	addd.2	|- ( ph -> B e. CC )
	addd.3	|- ( ph -> C e. CC )
	add4d.4	|- ( ph -> D e. CC )
Assertion	add42d	|- ( ph -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) )

Proof

Step	Hyp	Ref	Expression
1		addd.1	|- ( ph -> A e. CC )
2		addd.2	|- ( ph -> B e. CC )
3		addd.3	|- ( ph -> C e. CC )
4		add4d.4	|- ( ph -> 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	syl22anc	|- ( ph -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) )