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Theorem caov32 6275
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 6245 . . 3  |-  ( B F C )  =  ( C F B )
54oveq2i 6093 . 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 6248 . 2  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
96, 2, 1, 7caovass 6248 . 2  |-  ( ( A F C ) F B )  =  ( A F ( C F B ) )
105, 8, 93eqtr4i 2467 1  |-  ( ( A F B ) F C )  =  ( ( A F C ) F B )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   _Vcvv 2957  (class class class)co 6082
This theorem is referenced by:  caov31  6277  addassnq  8836  ltexprlem7  8920  mulcmpblnrlem  8949  recexsrlem  8979  mulgt0sr  8981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463  df-ov 6085
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