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Theorem caov32 5716
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
Hypotheses
Ref Expression
caov.1  |-  A  e. 
_V
caov.2  |-  B  e. 
_V
caov.3  |-  C  e. 
_V
caov.com  |-  ( x F y )  =  ( y F x )
caov.ass  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
Assertion
Ref Expression
caov32  |-  ( ( 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    x, F, y, z

Proof of Theorem caov32
StepHypRef Expression
1 caov.2 . . . 4  |-  B  e. 
_V
2 caov.3 . . . 4  |-  C  e. 
_V
3 caov.com . . . 4  |-  ( x F y )  =  ( y F x )
41, 2, 3caovcom 5686 . . 3  |-  ( B F C )  =  ( C F B )
54oveq2i 5551 . 2  |-  ( A F ( B F C ) )  =  ( A F ( C F B ) )
6 caov.1 . . 3  |-  A  e. 
_V
7 caov.ass . . 3  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
86, 1, 2, 7caovass 5689 . 2  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
96, 2, 1, 7caovass 5689 . 2  |-  ( ( A F C ) F B )  =  ( A F ( C F B ) )
105, 8, 93eqtr4i 2086 1  |-  ( ( A F B ) F C )  =  ( ( A F C ) F B )
Colors of variables: wff set class
Syntax hints:    = wceq 1259    e. wcel 1409   _Vcvv 2574  (class class class)co 5540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543
This theorem is referenced by:  caov31  5718
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