Metamath Proof Explorer

Theorem nnncan2

Description: Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005)

Ref Expression
Assertion nnncan2 
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - C ) - ( B - C ) ) = ( A - B ) )

Proof

Step Hyp Ref Expression
1 subcl 
 |-  ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
2 1 3adant1 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
3 sub32 
 |-  ( ( A e. CC /\ ( B - C ) e. CC /\ C e. CC ) -> ( ( A - ( B - C ) ) - C ) = ( ( A - C ) - ( B - C ) ) )
4 2 3 syld3an2 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - ( B - C ) ) - C ) = ( ( A - C ) - ( B - C ) ) )
5 nnncan 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - ( B - C ) ) - C ) = ( A - B ) )
6 4 5 eqtr3d 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - C ) - ( B - C ) ) = ( A - B ) )