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Theorem caov32d 5825
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
caov32d  |-  ( ph  ->  ( ( A F B ) F C )  =  ( ( A F C ) F B ) )
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 caov32d
StepHypRef Expression
1 caovd.com . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
2 caovd.2 . . . 4  |-  ( ph  ->  B  e.  S )
3 caovd.3 . . . 4  |-  ( ph  ->  C  e.  S )
41, 2, 3caovcomd 5801 . . 3  |-  ( ph  ->  ( B F C )  =  ( C F B ) )
54oveq2d 5668 . 2  |-  ( ph  ->  ( A F ( B F C ) )  =  ( A F ( C F B ) ) )
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.1 . . 3  |-  ( ph  ->  A  e.  S )
86, 7, 2, 3caovassd 5804 . 2  |-  ( ph  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
96, 7, 3, 2caovassd 5804 . 2  |-  ( ph  ->  ( ( A F C ) F B )  =  ( A F ( C F B ) ) )
105, 8, 93eqtr4d 2130 1  |-  ( ph  ->  ( ( A F B ) F C )  =  ( ( A F C ) F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    = wceq 1289    e. wcel 1438  (class class class)co 5652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655
This theorem is referenced by:  caov31d  5827  mulcanenq  6942  mulcanenq0ec  7002  ltexprlemrl  7167  ltexprlemru  7169  cauappcvgprlemladdfl  7212  cauappcvgprlemladdru  7213  mulcmpblnrlemg  7284  ltsosr  7308  recexgt0sr  7317  mulgt0sr  7321  caucvgsrlemoffcau  7341  caucvgsrlemoffres  7343  resqrexlemover  10439
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