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Theorem caov4 6280
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
caov4  |-  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( B F D ) )
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 caov4
StepHypRef Expression
1 caov.2 . . . 4  |-  B  e. 
_V
2 caov.3 . . . 4  |-  C  e. 
_V
3 caov.4 . . . 4  |-  D  e. 
_V
4 caov.com . . . 4  |-  ( x F y )  =  ( y F x )
5 caov.ass . . . 4  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
61, 2, 3, 4, 5caov12 6277 . . 3  |-  ( B F ( C F D ) )  =  ( C F ( B F D ) )
76oveq2i 6094 . 2  |-  ( A F ( B F ( C F D ) ) )  =  ( A F ( C F ( B F D ) ) )
8 caov.1 . . 3  |-  A  e. 
_V
9 ovex 6108 . . 3  |-  ( C F D )  e. 
_V
108, 1, 9, 5caovass 6249 . 2  |-  ( ( A F B ) F ( C F D ) )  =  ( A F ( B F ( C F D ) ) )
11 ovex 6108 . . 3  |-  ( B F D )  e. 
_V
128, 2, 11, 5caovass 6249 . 2  |-  ( ( A F C ) F ( B F D ) )  =  ( A F ( C F ( B F D ) ) )
137, 10, 123eqtr4i 2468 1  |-  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( B F D ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   _Vcvv 2958  (class class class)co 6083
This theorem is referenced by:  caov42  6282  ecopovtrn  7009  adderpqlem  8833  mulerpqlem  8834  ltmnq  8851  reclem3pr  8928  mulcmpblnrlem  8950  distrsr  8968  ltasr  8977  mulgt0sr  8982  axdistr  9035
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 2419  ax-nul 4340
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-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
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