Theorem addsubsub23

Description: Swap the second and the third terms in a difference of a sum and a difference (or, vice versa, in a sum of a difference and a sum). (Contributed by AV, 15-Nov-2025)

	Ref	Expression
Hypotheses	negidd.1	|- ( ph -> A e. CC )
	pncand.2	|- ( ph -> B e. CC )
	subaddd.3	|- ( ph -> C e. CC )
	addsub4d.4	|- ( ph -> D e. CC )
Assertion	addsubsub23	|- ( ph -> ( ( A + B ) - ( C - D ) ) = ( ( A - C ) + ( B + D ) ) )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	|- ( ph -> A e. CC )
2		pncand.2	|- ( ph -> B e. CC )
3		subaddd.3	|- ( ph -> C e. CC )
4		addsub4d.4	|- ( ph -> D e. CC )
5	1 2	addcld	|- ( ph -> ( A + B ) e. CC )
6	5 3 4	subsubd	|- ( ph -> ( ( A + B ) - ( C - D ) ) = ( ( ( A + B ) - C ) + D ) )
7	1 2 3	addsubd	|- ( ph -> ( ( A + B ) - C ) = ( ( A - C ) + B ) )
8	7	oveq1d	|- ( ph -> ( ( ( A + B ) - C ) + D ) = ( ( ( A - C ) + B ) + D ) )
9	1 3	subcld	|- ( ph -> ( A - C ) e. CC )
10	9 2 4	addassd	|- ( ph -> ( ( ( A - C ) + B ) + D ) = ( ( A - C ) + ( B + D ) ) )
11	6 8 10	3eqtrd	|- ( ph -> ( ( A + B ) - ( C - D ) ) = ( ( A - C ) + ( B + D ) ) )