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Theorem caov32d 6022
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 5998 . . 3  |-  ( ph  ->  ( B F C )  =  ( C F B ) )
54oveq2d 5858 . 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 6001 . 2  |-  ( ph  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
96, 7, 3, 2caovassd 6001 . 2  |-  ( ph  ->  ( ( A F C ) F B )  =  ( A F ( C F B ) ) )
105, 8, 93eqtr4d 2208 1  |-  ( ph  ->  ( ( A F B ) F C )  =  ( ( A F C ) F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 968    = wceq 1343    e. wcel 2136  (class class class)co 5842
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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845
This theorem is referenced by:  caov31d  6024  mulcanenq  7326  mulcanenq0ec  7386  ltexprlemrl  7551  ltexprlemru  7553  cauappcvgprlemladdfl  7596  cauappcvgprlemladdru  7597  mulcmpblnrlemg  7681  ltsosr  7705  recexgt0sr  7714  mulgt0sr  7719  caucvgsrlemoffcau  7739  caucvgsrlemoffres  7741  resqrexlemover  10952
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