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Theorem caov411 6408
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
caov411  |-  ( ( A F B ) F ( C F D ) )  =  ( ( C F B ) F ( A 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 caov411
StepHypRef Expression
1 caov.1 . . . 4  |-  A  e. 
_V
2 caov.2 . . . 4  |-  B  e. 
_V
3 caov.3 . . . 4  |-  C  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, 5caov31 6405 . . 3  |-  ( ( A F B ) F C )  =  ( ( C F B ) F A )
76oveq1i 6213 . 2  |-  ( ( ( A F B ) F C ) F D )  =  ( ( ( C F B ) F A ) F D )
8 ovex 6228 . . 3  |-  ( A F B )  e. 
_V
9 caov.4 . . 3  |-  D  e. 
_V
108, 3, 9, 5caovass 6376 . 2  |-  ( ( ( A F B ) F C ) F D )  =  ( ( A F B ) F ( C F D ) )
11 ovex 6228 . . 3  |-  ( C F B )  e. 
_V
1211, 1, 9, 5caovass 6376 . 2  |-  ( ( ( C F B ) F A ) F D )  =  ( ( C F B ) F ( A F D ) )
137, 10, 123eqtr3i 2491 1  |-  ( ( A F B ) F ( C F D ) )  =  ( ( C F B ) F ( A F D ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   _Vcvv 3078  (class class class)co 6203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4532
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-ov 6206
This theorem is referenced by:  ecopovtrn  7316  distrnq  9245  lterpq  9254  ltanq  9255  prlem936  9331
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