Theorem sltsubsub3bd

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

	Ref	Expression
Hypotheses	sltsubsubbd.1	|- ( ph -> A e. No )
	sltsubsubbd.2	|- ( ph -> B e. No )
	sltsubsubbd.3	|- ( ph -> C e. No )
	sltsubsubbd.4	|- ( ph -> D e. No )
Assertion	sltsubsub3bd	|- ( ph -> ( ( A -s C ) ( D -s C )

Proof

Step	Hyp	Ref	Expression
1		sltsubsubbd.1	|- ( ph -> A e. No )
2		sltsubsubbd.2	|- ( ph -> B e. No )
3		sltsubsubbd.3	|- ( ph -> C e. No )
4		sltsubsubbd.4	|- ( ph -> D e. No )
5	1 2 3 4	sltsubsubbd	|- ( ph -> ( ( A -s C ) ( A -s B )
6	1 2 3 4	sltsubsub2bd	|- ( ph -> ( ( A -s B ) ( D -s C )
7	5 6	bitrd	|- ( ph -> ( ( A -s C ) ( D -s C )