Theorem subeq0

Description: If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999)

	Ref	Expression
Assertion	subeq0	|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = 0 <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		subid	|- ( B e. CC -> ( B - B ) = 0 )
2	1	adantl	|- ( ( A e. CC /\ B e. CC ) -> ( B - B ) = 0 )
3	2	eqeq2d	|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = ( B - B ) <-> ( A - B ) = 0 ) )
4		subcan2	|- ( ( A e. CC /\ B e. CC /\ B e. CC ) -> ( ( A - B ) = ( B - B ) <-> A = B ) )
5	4	3anidm23	|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = ( B - B ) <-> A = B ) )
6	3 5	bitr3d	|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = 0 <-> A = B ) )