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Theorem caov12d 6078
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 6053 . . 3  |-  ( ph  ->  ( A F B )  =  ( B F A ) )
54oveq1d 5911 . 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 6056 . 2  |-  ( ph  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
96, 3, 2, 7caovassd 6056 . 2  |-  ( ph  ->  ( ( B F A ) F C )  =  ( B F ( A F C ) ) )
105, 8, 93eqtr3d 2230 1  |-  ( ph  ->  ( A F ( B F C ) )  =  ( B F ( A F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160  (class class class)co 5896
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5899
This theorem is referenced by:  caov4d  6081  caovimo  6090  ltaddnq  7436  ltexnqq  7437  enq0tr  7463  mullocprlem  7599  1idprl  7619  1idpru  7620  cauappcvgprlemdisj  7680  mulcmpblnrlemg  7769  lttrsr  7791  ltsosr  7793  0idsr  7796  1idsr  7797  recexgt0sr  7802  mulgt0sr  7807  axmulass  7902
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