Theorem lsubswap23d

Description: Swap the second and third variables in an equation with subtraction on the left, converting it into an addition.

EDITORIAL: The label for this theorem is questionable. Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 23-Aug-2024)

	Ref	Expression
Hypotheses	lsubswap23d.a	|- ( ph -> A e. CC )
	lsubswap23d.b	|- ( ph -> B e. CC )
	lsubswap23d.1	|- ( ph -> ( A - B ) = C )
Assertion	lsubswap23d	|- ( ph -> ( A - C ) = B )

Proof

Step	Hyp	Ref	Expression
1		lsubswap23d.a	|- ( ph -> A e. CC )
2		lsubswap23d.b	|- ( ph -> B e. CC )
3		lsubswap23d.1	|- ( ph -> ( A - B ) = C )
4	1 2	subcld	|- ( ph -> ( A - B ) e. CC )
5	3 4	eqeltrrd	|- ( ph -> C e. CC )
6	1 2 3	lsubrotld	|- ( ph -> ( B + C ) = A )
7	6	eqcomd	|- ( ph -> A = ( B + C ) )
8	2 5 7	mvrraddd	|- ( ph -> ( A - C ) = B )