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Theorem caov4d 6044
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 ) ) )
caovd.4  |-  ( ph  ->  D  e.  S )
caovd.cl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
Assertion
Ref Expression
caov4d  |-  ( ph  ->  ( ( 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    ph, x, y, z   
x, F, y, z   
x, S, y, z

Proof of Theorem caov4d
StepHypRef Expression
1 caovd.2 . . . 4  |-  ( ph  ->  B  e.  S )
2 caovd.3 . . . 4  |-  ( ph  ->  C  e.  S )
3 caovd.4 . . . 4  |-  ( ph  ->  D  e.  S )
4 caovd.com . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
5 caovd.ass . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
61, 2, 3, 4, 5caov12d 6041 . . 3  |-  ( ph  ->  ( B F ( C F D ) )  =  ( C F ( B F D ) ) )
76oveq2d 5874 . 2  |-  ( ph  ->  ( A F ( B F ( C F D ) ) )  =  ( A F ( C F ( B F D ) ) ) )
8 caovd.1 . . 3  |-  ( ph  ->  A  e.  S )
9 caovd.cl . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
109, 2, 3caovcld 6013 . . 3  |-  ( ph  ->  ( C F D )  e.  S )
115, 8, 1, 10caovassd 6019 . 2  |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( A F ( B F ( C F D ) ) ) )
129, 1, 3caovcld 6013 . . 3  |-  ( ph  ->  ( B F D )  e.  S )
135, 8, 2, 12caovassd 6019 . 2  |-  ( ph  ->  ( ( A F C ) F ( B F D ) )  =  ( A F ( C F ( B F D ) ) ) )
147, 11, 133eqtr4d 2325 1  |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( B F D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684  (class class class)co 5858
This theorem is referenced by:  caov411d  6045  caov42d  6046
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
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  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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