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Theorem caov12d 5920
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovd.1  |-  ( ph  ->  A  e.  S )
caovd.2  |-  ( ph  ->  B  e.  S )
caovd.3  |-  ( ph  ->  C  e.  S )
caovd.com  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
caovd.ass  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
Assertion
Ref Expression
caov12d  |-  ( ph  ->  ( A F ( B F C ) )  =  ( B F ( A F C ) ) )
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    ph, x, y, z   
x, F, y, z   
x, S, y, z

Proof of Theorem caov12d
StepHypRef Expression
1 caovd.com . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
2 caovd.1 . . . 4  |-  ( ph  ->  A  e.  S )
3 caovd.2 . . . 4  |-  ( ph  ->  B  e.  S )
41, 2, 3caovcomd 5895 . . 3  |-  ( ph  ->  ( A F B )  =  ( B F A ) )
54oveq1d 5757 . 2  |-  ( ph  ->  ( ( A F B ) F C )  =  ( ( B F A ) F C ) )
6 caovd.ass . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
7 caovd.3 . . 3  |-  ( ph  ->  C  e.  S )
86, 2, 3, 7caovassd 5898 . 2  |-  ( ph  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
96, 3, 2, 7caovassd 5898 . 2  |-  ( ph  ->  ( ( B F A ) F C )  =  ( B F ( A F C ) ) )
105, 8, 93eqtr3d 2158 1  |-  ( ph  ->  ( A F ( B F C ) )  =  ( B F ( A F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    = wceq 1316    e. wcel 1465  (class class class)co 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by:  caov4d  5923  caovimo  5932  ltaddnq  7183  ltexnqq  7184  enq0tr  7210  mullocprlem  7346  1idprl  7366  1idpru  7367  cauappcvgprlemdisj  7427  mulcmpblnrlemg  7516  lttrsr  7538  ltsosr  7540  0idsr  7543  1idsr  7544  recexgt0sr  7549  mulgt0sr  7554  axmulass  7649
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