Theorem add32r 7915

Description: Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)

Assertion

Ref	Expression
add32r	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( ( A + C ) + B ) )

Proof of Theorem add32r

Step	Hyp	Ref	Expression
1		addass 7743	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
2		add32 7914	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )
3	1, 2	eqtr3d 2172	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( ( A + C ) + B ) )

Syntax hints:   -> wi 4   /\ w3a 962   = wceq 1331   e. wcel 1480  (class class class)co 5767   CCcc 7611   + caddc 7616

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-addcom 7713  ax-addass 7715

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770

This theorem is referenced by:  bcxmas  11251