Theorem addsubs4d

Description: Rearrangement of four terms in mixed addition and subtraction. Surreal version. (Contributed by Scott Fenton, 25-Jul-2025)

	Ref	Expression
Hypotheses	addsubs4d.1	|- ( ph -> A e. No )
	addsubs4d.2	|- ( ph -> B e. No )
	addsubs4d.3	|- ( ph -> C e. No )
	addsubs4d.4	|- ( ph -> D e. No )
Assertion	addsubs4d	|- ( ph -> ( ( A +s B ) -s ( C +s D ) ) = ( ( A -s C ) +s ( B -s D ) ) )

Proof

Step	Hyp	Ref	Expression
1		addsubs4d.1	|- ( ph -> A e. No )
2		addsubs4d.2	|- ( ph -> B e. No )
3		addsubs4d.3	|- ( ph -> C e. No )
4		addsubs4d.4	|- ( ph -> D e. No )
5	1 2 3	addsubsd	|- ( ph -> ( ( A +s B ) -s C ) = ( ( A -s C ) +s B ) )
6	5	oveq1d	|- ( ph -> ( ( ( A +s B ) -s C ) -s D ) = ( ( ( A -s C ) +s B ) -s D ) )
7	1 2	addscld	|- ( ph -> ( A +s B ) e. No )
8	7 3 4	subsubs4d	|- ( ph -> ( ( ( A +s B ) -s C ) -s D ) = ( ( A +s B ) -s ( C +s D ) ) )
9	1 3	subscld	|- ( ph -> ( A -s C ) e. No )
10	9 2 4	addsubsassd	|- ( ph -> ( ( ( A -s C ) +s B ) -s D ) = ( ( A -s C ) +s ( B -s D ) ) )
11	6 8 10	3eqtr3d	|- ( ph -> ( ( A +s B ) -s ( C +s D ) ) = ( ( A -s C ) +s ( B -s D ) ) )