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Theorem caov13 6244
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
caov13  |-  ( A F ( B F C ) )  =  ( C F ( B F A ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    x, F, y, z

Proof of Theorem caov13
StepHypRef Expression
1 caov.1 . . 3  |-  A  e. 
_V
2 caov.2 . . 3  |-  B  e. 
_V
3 caov.3 . . 3  |-  C  e. 
_V
4 caov.com . . 3  |-  ( x F y )  =  ( y F x )
5 caov.ass . . 3  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
61, 2, 3, 4, 5caov31 6243 . 2  |-  ( ( A F B ) F C )  =  ( ( C F B ) F A )
71, 2, 3, 5caovass 6214 . 2  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
83, 2, 1, 5caovass 6214 . 2  |-  ( ( C F B ) F A )  =  ( C F ( B F A ) )
96, 7, 83eqtr3i 2440 1  |-  ( A F ( B F C ) )  =  ( C F ( B F A ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   _Vcvv 2924  (class class class)co 6048
This theorem is referenced by:  ltsonq  8810  mulcmpblnrlem  8912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-ov 6051
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