Theorem subaddsd

Description: Relationship between addition and subtraction for surreals. (Contributed by Scott Fenton, 5-Feb-2025)

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

Proof

Step	Hyp	Ref	Expression
1		subaddsd.1	|- ( ph -> A e. No )
2		subaddsd.2	|- ( ph -> B e. No )
3		subaddsd.3	|- ( ph -> C e. No )
4		subadds	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A -s B ) = C <-> ( B +s C ) = A ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( ( A -s B ) = C <-> ( B +s C ) = A ) )