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Theorem add32i 8040
Description: Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
Hypotheses
Ref Expression
add.1  |-  A  e.  CC
add.2  |-  B  e.  CC
add.3  |-  C  e.  CC
Assertion
Ref Expression
add32i  |-  ( ( A  +  B )  +  C )  =  ( ( A  +  C )  +  B
)

Proof of Theorem add32i
StepHypRef Expression
1 add.1 . 2  |-  A  e.  CC
2 add.2 . 2  |-  B  e.  CC
3 add.3 . 2  |-  C  e.  CC
4 add32 8035 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( ( A  +  C )  +  B ) )
51, 2, 3, 4mp3an 1319 1  |-  ( ( A  +  B )  +  C )  =  ( ( A  +  C )  +  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1335    e. wcel 2128  (class class class)co 5825   CCcc 7731    + caddc 7736
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-addcom 7833  ax-addass 7835
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-iota 5136  df-fv 5179  df-ov 5828
This theorem is referenced by: (None)
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