Theorem int-addcomd

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

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

Proof

Step	Hyp	Ref	Expression
1		int-addcomd.1	|- ( ph -> B e. RR )
2		int-addcomd.2	|- ( ph -> C e. RR )
3		int-addcomd.3	|- ( ph -> A = B )
4	1	recnd	|- ( ph -> B e. CC )
5	2	recnd	|- ( ph -> C e. CC )
6	4 5	addcomd	|- ( ph -> ( B + C ) = ( C + B ) )
7	3	eqcomd	|- ( ph -> B = A )
8	7	oveq2d	|- ( ph -> ( C + B ) = ( C + A ) )
9	6 8	eqtrd	|- ( ph -> ( B + C ) = ( C + A ) )