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Theorem caov12d 5710
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 5685 . . 3  |-  ( ph  ->  ( A F B )  =  ( B F A ) )
54oveq1d 5555 . 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 5688 . 2  |-  ( ph  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
96, 3, 2, 7caovassd 5688 . 2  |-  ( ph  ->  ( ( B F A ) F C )  =  ( B F ( A F C ) ) )
105, 8, 93eqtr3d 2096 1  |-  ( ph  ->  ( A F ( B F C ) )  =  ( B F ( A F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    /\ w3a 896    = wceq 1259    e. wcel 1409  (class class class)co 5540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543
This theorem is referenced by:  caov4d  5713  caovimo  5722  ltaddnq  6563  ltexnqq  6564  enq0tr  6590  mullocprlem  6726  1idprl  6746  1idpru  6747  cauappcvgprlemdisj  6807  mulcmpblnrlemg  6883  lttrsr  6905  ltsosr  6907  0idsr  6910  1idsr  6911  recexgt0sr  6916  mulgt0sr  6920  axmulass  7005
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