Theorem mul12i 7908

Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Hypotheses

Ref	Expression
mul.1	|- A e. CC
mul.2	|- B e. CC
mul.3	|- C e. CC

Assertion

Ref	Expression
mul12i	|- ( A x. ( B x. C ) ) = ( B x. ( A x. C ) )

Proof of Theorem mul12i

Step	Hyp	Ref	Expression
1		mul.1	. 2 |- A e. CC
2		mul.2	. 2 |- B e. CC
3		mul.3	. 2 |- C e. CC
4		mul12 7891	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )
5	1, 2, 3, 4	mp3an 1315	1 |- ( A x. ( B x. C ) ) = ( B x. ( A x. C ) )

Syntax hints:   = wceq 1331   e. wcel 1480  (class class class)co 5774   CCcc 7618   x. cmul 7625

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 2121  ax-mulcom 7721  ax-mulass 7723

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777

This theorem is referenced by:  decmul10add  9250