Theorem reladdrsub

Description: Move LHS of a sum into RHS of a (real) difference. Version of mvlladdd with real subtraction. (Contributed by Steven Nguyen, 8-Jan-2023)

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

Proof

Step	Hyp	Ref	Expression
1		reladdrsub.1	|- ( ph -> A e. RR )
2		reladdrsub.2	|- ( ph -> B e. RR )
3		reladdrsub.3	|- ( ph -> ( A + B ) = C )
4	1 2	readdcld	|- ( ph -> ( A + B ) e. RR )
5	3 4	eqeltrrd	|- ( ph -> C e. RR )
6		resubadd	|- ( ( C e. RR /\ A e. RR /\ B e. RR ) -> ( ( C -R A ) = B <-> ( A + B ) = C ) )
7	3 6	syl5ibrcom	|- ( ph -> ( ( C e. RR /\ A e. RR /\ B e. RR ) -> ( C -R A ) = B ) )
8	5 1 2 7	mp3and	|- ( ph -> ( C -R A ) = B )
9	8	eqcomd	|- ( ph -> B = ( C -R A ) )