Theorem subeq0d

Description: If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	negidd.1	|- ( ph -> A e. CC )
	pncand.2	|- ( ph -> B e. CC )
	subeq0d.3	|- ( ph -> ( A - B ) = 0 )
Assertion	subeq0d	|- ( ph -> A = B )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	|- ( ph -> A e. CC )
2		pncand.2	|- ( ph -> B e. CC )
3		subeq0d.3	|- ( ph -> ( A - B ) = 0 )
4		subeq0	|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = 0 <-> A = B ) )
5	1 2 4	syl2anc	|- ( ph -> ( ( A - B ) = 0 <-> A = B ) )
6	3 5	mpbid	|- ( ph -> A = B )