Theorem subsub

Description: Law for double subtraction. (Contributed by NM, 13-May-2004)

	Ref	Expression
Assertion	subsub	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( ( A - B ) + C ) )

Proof

Step	Hyp	Ref	Expression
1		subsub2	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( A + ( C - B ) ) )
2		addsubass	|- ( ( A e. CC /\ C e. CC /\ B e. CC ) -> ( ( A + C ) - B ) = ( A + ( C - B ) ) )
3		addsub	|- ( ( A e. CC /\ C e. CC /\ B e. CC ) -> ( ( A + C ) - B ) = ( ( A - B ) + C ) )
4	2 3	eqtr3d	|- ( ( A e. CC /\ C e. CC /\ B e. CC ) -> ( A + ( C - B ) ) = ( ( A - B ) + C ) )
5	4	3com23	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( C - B ) ) = ( ( A - B ) + C ) )
6	1 5	eqtrd	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( ( A - B ) + C ) )