Theorem int-addassocd

Description: AdditionAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-addassocd.1	|- ( ph -> A e. RR )
	int-addassocd.2	|- ( ph -> C e. RR )
	int-addassocd.3	|- ( ph -> D e. RR )
	int-addassocd.4	|- ( ph -> A = B )
Assertion	int-addassocd	|- ( ph -> ( B + ( C + D ) ) = ( ( A + C ) + D ) )

Proof

Step	Hyp	Ref	Expression
1		int-addassocd.1	|- ( ph -> A e. RR )
2		int-addassocd.2	|- ( ph -> C e. RR )
3		int-addassocd.3	|- ( ph -> D e. RR )
4		int-addassocd.4	|- ( ph -> A = B )
5	1	recnd	|- ( ph -> A e. CC )
6	2	recnd	|- ( ph -> C e. CC )
7	3	recnd	|- ( ph -> D e. CC )
8	5 6 7	addassd	|- ( ph -> ( ( A + C ) + D ) = ( A + ( C + D ) ) )
9	4	oveq1d	|- ( ph -> ( A + ( C + D ) ) = ( B + ( C + D ) ) )
10	8 9	eqtr2d	|- ( ph -> ( B + ( C + D ) ) = ( ( A + C ) + D ) )