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Theorem caov32 6393
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 ) )
Assertion
Ref Expression
caov32  |-  ( ( A F B ) F C )  =  ( ( A F C ) F B )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    x, F, y, z

Proof of Theorem caov32
StepHypRef Expression
1 caov.2 . . . 4  |-  B  e. 
_V
2 caov.3 . . . 4  |-  C  e. 
_V
3 caov.com . . . 4  |-  ( x F y )  =  ( y F x )
41, 2, 3caovcom 6363 . . 3  |-  ( B F C )  =  ( C F B )
54oveq2i 6204 . 2  |-  ( A F ( B F C ) )  =  ( A F ( C F B ) )
6 caov.1 . . 3  |-  A  e. 
_V
7 caov.ass . . 3  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
86, 1, 2, 7caovass 6366 . 2  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
96, 2, 1, 7caovass 6366 . 2  |-  ( ( A F C ) F B )  =  ( A F ( C F B ) )
105, 8, 93eqtr4i 2490 1  |-  ( ( A F B ) F C )  =  ( ( A F C ) F B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   _Vcvv 3071  (class class class)co 6193
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 1952  ax-ext 2430
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-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-ov 6196
This theorem is referenced by:  caov31  6395  addassnq  9231  ltexprlem7  9315  mulcmpblnrlem  9344  recexsrlem  9374  mulgt0sr  9376
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