Theorem subsubadd23

Description: Swap the second and the third terms in a difference 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	subsubadd23	|- ( 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 3	sub32d	|- ( ph -> ( ( A - B ) - C ) = ( ( A - C ) - B ) )
6	5	oveq1d	|- ( ph -> ( ( ( A - B ) - C ) - D ) = ( ( ( A - C ) - B ) - D ) )
7	1 2	subcld	|- ( ph -> ( A - B ) e. CC )
8	7 3 4	subsub4d	|- ( ph -> ( ( ( A - B ) - C ) - D ) = ( ( A - B ) - ( C + D ) ) )
9	1 3	subcld	|- ( ph -> ( A - C ) e. CC )
10	9 2 4	subsub4d	|- ( ph -> ( ( ( A - C ) - B ) - D ) = ( ( A - C ) - ( B + D ) ) )
11	6 8 10	3eqtr3d	|- ( ph -> ( ( A - B ) - ( C + D ) ) = ( ( A - C ) - ( B + D ) ) )