Theorem mvrrsubd

Description: Move a subtraction in the RHS to a right-addition in the LHS. Converse of mvlraddd . (Contributed by SN, 21-Aug-2024)

	Ref	Expression
Hypotheses	mvrrsubd.a	|- ( ph -> B e. CC )
	mvrrsubd.b	|- ( ph -> C e. CC )
	mvrrsubd.1	|- ( ph -> A = ( B - C ) )
Assertion	mvrrsubd	|- ( ph -> ( A + C ) = B )

Proof

Step	Hyp	Ref	Expression
1		mvrrsubd.a	|- ( ph -> B e. CC )
2		mvrrsubd.b	|- ( ph -> C e. CC )
3		mvrrsubd.1	|- ( ph -> A = ( B - C ) )
4	1 2	subcld	|- ( ph -> ( B - C ) e. CC )
5	3 4	eqeltrd	|- ( ph -> A e. CC )
6	5 2	addcld	|- ( ph -> ( A + C ) e. CC )
7	5 2	pncand	|- ( ph -> ( ( A + C ) - C ) = A )
8	7 3	eqtrd	|- ( ph -> ( ( A + C ) - C ) = ( B - C ) )
9	6 1 2 8	subcan2d	|- ( ph -> ( A + C ) = B )