Theorem ltsubsubs3bd

Description: Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 21-Feb-2025)

	Ref	Expression
Hypotheses	ltsubsubsbd.1	|- ( ph -> A e. No )
	ltsubsubsbd.2	|- ( ph -> B e. No )
	ltsubsubsbd.3	|- ( ph -> C e. No )
	ltsubsubsbd.4	|- ( ph -> D e. No )
Assertion	ltsubsubs3bd	|- ( ph -> ( ( A -s C ) ( D -s C )

Proof

Step	Hyp	Ref	Expression
1		ltsubsubsbd.1	|- ( ph -> A e. No )
2		ltsubsubsbd.2	|- ( ph -> B e. No )
3		ltsubsubsbd.3	|- ( ph -> C e. No )
4		ltsubsubsbd.4	|- ( ph -> D e. No )
5	1 2 3 4	ltsubsubsbd	|- ( ph -> ( ( A -s C ) ( A -s B )
6	1 2 3 4	ltsubsubs2bd	|- ( ph -> ( ( A -s B ) ( D -s C )
7	5 6	bitrd	|- ( ph -> ( ( A -s C ) ( D -s C )