Theorem subadd23

Description: Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007)

	Ref	Expression
Assertion	subadd23	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + C ) = ( A + ( C - B ) ) )

Proof

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