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Theorem caov4 6051
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 6048 . . 3  |-  ( B F ( C F D ) )  =  ( C F ( B F D ) )
76oveq2i 5869 . 2  |-  ( A F ( B F ( C F D ) ) )  =  ( A F ( C F ( B F D ) ) )
8 caov.1 . . 3  |-  A  e. 
_V
9 ovex 5883 . . 3  |-  ( C F D )  e. 
_V
108, 1, 9, 5caovass 6020 . 2  |-  ( ( A F B ) F ( C F D ) )  =  ( A F ( B F ( C F D ) ) )
11 ovex 5883 . . 3  |-  ( B F D )  e. 
_V
128, 2, 11, 5caovass 6020 . 2  |-  ( ( A F C ) F ( B F D ) )  =  ( A F ( C F ( B F D ) ) )
137, 10, 123eqtr4i 2313 1  |-  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( B F D ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788  (class class class)co 5858
This theorem is referenced by:  caov42  6053  ecopovtrn  6761  adderpqlem  8578  mulerpqlem  8579  ltmnq  8596  reclem3pr  8673  mulcmpblnrlem  8695  distrsr  8713  ltasr  8722  mulgt0sr  8727  axdistr  8780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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