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Theorem caov32d 5944
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 5920 . . 3  |-  ( ph  ->  ( B F C )  =  ( C F B ) )
54oveq2d 5783 . 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 5923 . 2  |-  ( ph  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
96, 7, 3, 2caovassd 5923 . 2  |-  ( ph  ->  ( ( A F C ) F B )  =  ( A F ( C F B ) ) )
105, 8, 93eqtr4d 2180 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 962    = wceq 1331    e. wcel 1480  (class class class)co 5767
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770
This theorem is referenced by:  caov31d  5946  mulcanenq  7186  mulcanenq0ec  7246  ltexprlemrl  7411  ltexprlemru  7413  cauappcvgprlemladdfl  7456  cauappcvgprlemladdru  7457  mulcmpblnrlemg  7541  ltsosr  7565  recexgt0sr  7574  mulgt0sr  7579  caucvgsrlemoffcau  7599  caucvgsrlemoffres  7601  resqrexlemover  10775
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