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Theorem caov4d 6060
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 6057 . . 3  |-  ( ph  ->  ( B F ( C F D ) )  =  ( C F ( B F D ) ) )
76oveq2d 5890 . 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 6029 . . 3  |-  ( ph  ->  ( C F D )  e.  S )
115, 8, 1, 10caovassd 6035 . 2  |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( A F ( B F ( C F D ) ) ) )
129, 1, 3caovcld 6029 . . 3  |-  ( ph  ->  ( B F D )  e.  S )
135, 8, 2, 12caovassd 6035 . 2  |-  ( ph  ->  ( ( A F C ) F ( B F D ) )  =  ( A F ( C F ( B F D ) ) ) )
147, 11, 133eqtr4d 2338 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 1632    e. wcel 1696  (class class class)co 5874
This theorem is referenced by:  caov411d  6061  caov42d  6062
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
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  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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