Theorem add12 8117

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 8096	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
2	1	oveq1d 5892	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + C ) = ( ( B + A ) + C ) )
3	2	3adant3 1017	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( B + A ) + C ) )
4		addass 7943	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
5		addass 7943	. . 3 |- ( ( B e. CC /\ A e. CC /\ C e. CC ) -> ( ( B + A ) + C ) = ( B + ( A + C ) ) )
6	5	3com12 1207	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B + A ) + C ) = ( B + ( A + C ) ) )
7	3, 4, 6	3eqtr3d 2218	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( B + ( A + C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148  (class class class)co 5877   CCcc 7811   + caddc 7816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-addcom 7913  ax-addass 7915

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880

This theorem is referenced by:  add4  8120  add12i  8122  add12d  8126