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Theorem add12 8145
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
StepHypRef Expression
1 addcom 8124 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
21oveq1d 5911 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  C
)  =  ( ( B  +  A )  +  C ) )
323adant3 1019 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( ( B  +  A )  +  C ) )
4 addass 7971 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( A  +  ( B  +  C
) ) )
5 addass 7971 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  (
( B  +  A
)  +  C )  =  ( B  +  ( A  +  C
) ) )
653com12 1209 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  +  A
)  +  C )  =  ( B  +  ( A  +  C
) ) )
73, 4, 63eqtr3d 2230 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  +  C ) )  =  ( B  +  ( A  +  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160  (class class class)co 5896   CCcc 7839    + caddc 7844
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-addcom 7941  ax-addass 7943
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5899
This theorem is referenced by:  add4  8148  add12i  8150  add12d  8154
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