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Metamath Proof Explorer

Theorem pnpcan2

Description: Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006)

Ref Expression
Assertion pnpcan2 
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - ( B + C ) ) = ( A - B ) )

Proof

Step Hyp Ref Expression
1 addcom 
 |-  ( ( A e. CC /\ C e. CC ) -> ( A + C ) = ( C + A ) )
2 1 3adant2 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + C ) = ( C + A ) )
3 addcom 
 |-  ( ( B e. CC /\ C e. CC ) -> ( B + C ) = ( C + B ) )
4 3 3adant1 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B + C ) = ( C + B ) )
5 2 4 oveq12d 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - ( B + C ) ) = ( ( C + A ) - ( C + B ) ) )
6 pnpcan 
 |-  ( ( C e. CC /\ A e. CC /\ B e. CC ) -> ( ( C + A ) - ( C + B ) ) = ( A - B ) )
7 6 3coml 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( C + A ) - ( C + B ) ) = ( A - B ) )
8 5 7 eqtrd 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - ( B + C ) ) = ( A - B ) )