Theorem add12 8056

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

Assertion

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

Proof of Theorem add12

Step	Hyp	Ref	Expression
1		addcom 8035	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
2	1	oveq1d 5857	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + C ) = ( ( B + A ) + C ) )
3	2	3adant3 1007	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( B + A ) + C ) )
4		addass 7883	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
5		addass 7883	. . 3 |- ( ( B e. CC /\ A e. CC /\ C e. CC ) -> ( ( B + A ) + C ) = ( B + ( A + C ) ) )
6	5	3com12 1197	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B + A ) + C ) = ( B + ( A + C ) ) )
7	3, 4, 6	3eqtr3d 2206	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( B + ( A + C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136  (class class class)co 5842   CCcc 7751   + caddc 7756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-addcom 7853  ax-addass 7855

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  add4  8059  add12i  8061  add12d  8065