Theorem add12d

Description: Commutative/associative law that swaps the first two terms in a triple 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 )
Assertion	add12d	|- ( ph -> ( A + ( B + C ) ) = ( B + ( A + C ) ) )

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		add12	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( B + ( A + C ) ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( A + ( B + C ) ) = ( B + ( A + C ) ) )