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Theorem caov42 6095
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 ) )
caov.4  |-  D  e. 
_V
Assertion
Ref Expression
caov42  |-  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( D F B ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    x, D, y, z    x, F, y, z

Proof of Theorem caov42
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 ) )
6 caov.4 . . 3  |-  D  e. 
_V
71, 2, 3, 4, 5, 6caov4 6093 . 2  |-  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( B F D ) )
82, 6, 4caovcom 6059 . . 3  |-  ( B F D )  =  ( D F B )
98oveq2i 5911 . 2  |-  ( ( A F C ) F ( B F D ) )  =  ( ( A F C ) F ( D F B ) )
107, 9eqtri 2336 1  |-  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( D F B ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1633    e. wcel 1701   _Vcvv 2822  (class class class)co 5900
This theorem is referenced by:  caovlem2  6098  mulcmpblnrlem  8740  ltasr  8767  axmulass  8824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-nul 4186
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-ov 5903
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