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Theorem caov411 6444
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 6441 . . 3  |-  ( ( A F B ) F C )  =  ( ( C F B ) F A )
76oveq1i 6244 . 2  |-  ( ( ( A F B ) F C ) F D )  =  ( ( ( C F B ) F A ) F D )
8 ovex 6262 . . 3  |-  ( A F B )  e. 
_V
9 caov.4 . . 3  |-  D  e. 
_V
108, 3, 9, 5caovass 6412 . 2  |-  ( ( ( A F B ) F C ) F D )  =  ( ( A F B ) F ( C F D ) )
11 ovex 6262 . . 3  |-  ( C F B )  e. 
_V
1211, 1, 9, 5caovass 6412 . 2  |-  ( ( ( C F B ) F A ) F D )  =  ( ( C F B ) F ( A F D ) )
137, 10, 123eqtr3i 2439 1  |-  ( ( A F B ) F ( C F D ) )  =  ( ( C F B ) F ( A F D ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    e. wcel 1842   _Vcvv 3058  (class class class)co 6234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5489  df-fv 5533  df-ov 6237
This theorem is referenced by:  ecopovtrn  7371  distrnq  9289  lterpq  9298  ltanq  9299  prlem936  9375
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