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Theorem caov4 6067
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 6064 . . 3  |-  ( B F ( C F D ) )  =  ( C F ( B F D ) )
76oveq2i 5885 . 2  |-  ( A F ( B F ( C F D ) ) )  =  ( A F ( C F ( B F D ) ) )
8 caov.1 . . 3  |-  A  e. 
_V
9 ovex 5899 . . 3  |-  ( C F D )  e. 
_V
108, 1, 9, 5caovass 6036 . 2  |-  ( ( A F B ) F ( C F D ) )  =  ( A F ( B F ( C F D ) ) )
11 ovex 5899 . . 3  |-  ( B F D )  e. 
_V
128, 2, 11, 5caovass 6036 . 2  |-  ( ( A F C ) F ( B F D ) )  =  ( A F ( C F ( B F D ) ) )
137, 10, 123eqtr4i 2326 1  |-  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( B F D ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801  (class class class)co 5874
This theorem is referenced by:  caov42  6069  ecopovtrn  6777  adderpqlem  8594  mulerpqlem  8595  ltmnq  8612  reclem3pr  8689  mulcmpblnrlem  8711  distrsr  8729  ltasr  8738  mulgt0sr  8743  axdistr  8796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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