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Theorem caov12d 6261
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovd.1  |-  ( ph  ->  A  e.  S )
caovd.2  |-  ( ph  ->  B  e.  S )
caovd.3  |-  ( ph  ->  C  e.  S )
caovd.com  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
caovd.ass  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
Assertion
Ref Expression
caov12d  |-  ( ph  ->  ( A F ( B F C ) )  =  ( B F ( A F C ) ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    ph, x, y, z   
x, F, y, z   
x, S, y, z

Proof of Theorem caov12d
StepHypRef Expression
1 caovd.com . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
2 caovd.1 . . . 4  |-  ( ph  ->  A  e.  S )
3 caovd.2 . . . 4  |-  ( ph  ->  B  e.  S )
41, 2, 3caovcomd 6236 . . 3  |-  ( ph  ->  ( A F B )  =  ( B F A ) )
54oveq1d 6089 . 2  |-  ( ph  ->  ( ( A F B ) F C )  =  ( ( B F A ) F C ) )
6 caovd.ass . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
7 caovd.3 . . 3  |-  ( ph  ->  C  e.  S )
86, 2, 3, 7caovassd 6239 . 2  |-  ( ph  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
96, 3, 2, 7caovassd 6239 . 2  |-  ( ph  ->  ( ( B F A ) F C )  =  ( B F ( A F C ) ) )
105, 8, 93eqtr3d 2476 1  |-  ( ph  ->  ( A F ( B F C ) )  =  ( B F ( A F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725  (class class class)co 6074
This theorem is referenced by:  caov4d  6264  psrass23  16466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-iota 5411  df-fv 5455  df-ov 6077
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