Theorem lsubcom23d

Description: Swap the second and third variables in an equation with subtraction on the left, converting it into an addition. (Contributed by SN, 23-Aug-2024)

	Ref	Expression
Hypotheses	lsubcom23d.a	|- ( ph -> A e. CC )
	lsubcom23d.b	|- ( ph -> B e. CC )
	lsubcom23d.1	|- ( ph -> ( A - B ) = C )
Assertion	lsubcom23d	|- ( ph -> ( A - C ) = B )

Proof

Step	Hyp	Ref	Expression
1		lsubcom23d.a	|- ( ph -> A e. CC )
2		lsubcom23d.b	|- ( ph -> B e. CC )
3		lsubcom23d.1	|- ( ph -> ( A - B ) = C )
4	1 2	subcld	|- ( ph -> ( A - B ) e. CC )
5	3 4	eqeltrrd	|- ( ph -> C e. CC )
6	1 5	subcld	|- ( ph -> ( A - C ) e. CC )
7	4 3	subeq0bd	|- ( ph -> ( ( A - B ) - C ) = 0 )
8	1 5 2	sub32d	|- ( ph -> ( ( A - C ) - B ) = ( ( A - B ) - C ) )
9	2	subidd	|- ( ph -> ( B - B ) = 0 )
10	7 8 9	3eqtr4d	|- ( ph -> ( ( A - C ) - B ) = ( B - B ) )
11	6 2 2 10	subcan2d	|- ( ph -> ( A - C ) = B )