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Theorem caov42d 6213
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
caov42d  |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( D F B ) ) )
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 caov42d
StepHypRef Expression
1 caovd.1 . . 3  |-  ( ph  ->  A  e.  S )
2 caovd.2 . . 3  |-  ( ph  ->  B  e.  S )
3 caovd.3 . . 3  |-  ( ph  ->  C  e.  S )
4 caovd.com . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
5 caovd.ass . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
6 caovd.4 . . 3  |-  ( ph  ->  D  e.  S )
7 caovd.cl . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
81, 2, 3, 4, 5, 6, 7caov4d 6211 . 2  |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( B F D ) ) )
94, 2, 6caovcomd 6183 . . 3  |-  ( ph  ->  ( B F D )  =  ( D F B ) )
109oveq2d 6037 . 2  |-  ( ph  ->  ( ( A F C ) F ( B F D ) )  =  ( ( A F C ) F ( D F B ) ) )
118, 10eqtrd 2420 1  |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( D F B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717  (class class class)co 6021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-ov 6024
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