Theorem add32i 8058

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

Step	Hyp	Ref	Expression
1		add.1	. 2 |- A e. CC
2		add.2	. 2 |- B e. CC
3		add.3	. 2 |- C e. CC
4		add32 8053	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )
5	1, 2, 3, 4	mp3an 1327	1 |- ( ( A + B ) + C ) = ( ( A + C ) + B )

Syntax hints:   = wceq 1343   e. wcel 2136  (class class class)co 5841   CCcc 7747   + caddc 7752

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 7849  ax-addass 7851

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 2296  df-rex 2449  df-v 2727  df-un 3119  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-br 3982  df-iota 5152  df-fv 5195  df-ov 5844

This theorem is referenced by:  lgsdir2lem2  13530