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Metamath Proof Explorer

Theorem add12

Description: Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004)

Ref Expression
Assertion add12 
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( B + ( A + C ) ) )

Proof

Step Hyp Ref Expression
1 addcom 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
2 1 oveq1d 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + C ) = ( ( B + A ) + C ) )
3 2 3adant3 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( B + A ) + C ) )
4 addass 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
5 addass 
 |-  ( ( B e. CC /\ A e. CC /\ C e. CC ) -> ( ( B + A ) + C ) = ( B + ( A + C ) ) )
6 5 3com12 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B + A ) + C ) = ( B + ( A + C ) ) )
7 3 4 6 3eqtr3d 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( B + ( A + C ) ) )