Theorem subsubs2d

Description: Law for double surreal subtraction. (Contributed by Scott Fenton, 16-Apr-2025)

	Ref	Expression
Hypotheses	subsubs4d.1	|- ( ph -> A e. No )
	subsubs4d.2	|- ( ph -> B e. No )
	subsubs4d.3	|- ( ph -> C e. No )
Assertion	subsubs2d	|- ( ph -> ( A -s ( B -s C ) ) = ( A +s ( C -s B ) ) )

Proof

Step	Hyp	Ref	Expression
1		subsubs4d.1	|- ( ph -> A e. No )
2		subsubs4d.2	|- ( ph -> B e. No )
3		subsubs4d.3	|- ( ph -> C e. No )
4	2 3	subscld	|- ( ph -> ( B -s C ) e. No )
5	1 4	subsvald	|- ( ph -> ( A -s ( B -s C ) ) = ( A +s ( -us ` ( B -s C ) ) ) )
6	2 3	negsubsdi2d	|- ( ph -> ( -us ` ( B -s C ) ) = ( C -s B ) )
7	6	oveq2d	|- ( ph -> ( A +s ( -us ` ( B -s C ) ) ) = ( A +s ( C -s B ) ) )
8	5 7	eqtrd	|- ( ph -> ( A -s ( B -s C ) ) = ( A +s ( C -s B ) ) )