Theorem mul12i 8384

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 8367	. 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 1374	1 |- ( A x. ( B x. C ) ) = ( B x. ( A x. C ) )

Syntax hints:   = wceq 1398   e. wcel 2202  (class class class)co 6028   CCcc 8090   x. cmul 8097

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-mulcom 8193  ax-mulass 8195

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  decmul10add  9740  decsplit  13082