Theorem subsub3

Description: Law for double subtraction. (Contributed by NM, 27-Jul-2005)

	Ref	Expression
Assertion	subsub3	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( ( A + C ) - B ) )

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	2	3com23	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - B ) = ( A + ( C - B ) ) )
4	1 3	eqtr4d	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( ( A + C ) - B ) )