Theorem addscan1d

Description: Cancellation law for surreal addition. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypotheses	addscand.1	|- ( ph -> A e. No )
	addscand.2	|- ( ph -> B e. No )
	addscand.3	|- ( ph -> C e. No )
Assertion	addscan1d	|- ( ph -> ( ( C +s A ) = ( C +s B ) <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		addscand.1	|- ( ph -> A e. No )
2		addscand.2	|- ( ph -> B e. No )
3		addscand.3	|- ( ph -> C e. No )
4		addscan1	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( C +s A ) = ( C +s B ) <-> A = B ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( ( C +s A ) = ( C +s B ) <-> A = B ) )