ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caov42d Unicode version

Theorem caov42d 5847
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 5845 . 2  |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( B F D ) ) )
94, 2, 6caovcomd 5817 . . 3  |-  ( ph  ->  ( B F D )  =  ( D F B ) )
109oveq2d 5684 . 2  |-  ( ph  ->  ( ( A F C ) F ( B F D ) )  =  ( ( A F C ) F ( D F B ) ) )
118, 10eqtrd 2121 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 103    /\ w3a 925    = wceq 1290    e. wcel 1439  (class class class)co 5668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-iota 4995  df-fv 5038  df-ov 5671
This theorem is referenced by:  caovlem2d  5853  mulcmpblnrlemg  7349  ltasrg  7379  axmulass  7471
  Copyright terms: Public domain W3C validator