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Theorem caovass 6053
Description: Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
Hypotheses
Ref Expression
caovass.1  |-  A  e. 
_V
caovass.2  |-  B  e. 
_V
caovass.3  |-  C  e. 
_V
caovass.4  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
Assertion
Ref Expression
caovass  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    x, F, y, z

Proof of Theorem caovass
StepHypRef Expression
1 caovass.1 . 2  |-  A  e. 
_V
2 caovass.2 . 2  |-  B  e. 
_V
3 caovass.3 . 2  |-  C  e. 
_V
4 tru 1368 . . 3  |- T.
5 caovass.4 . . . . 5  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
65a1i 9 . . . 4  |-  ( ( T.  /\  ( x  e.  _V  /\  y  e.  _V  /\  z  e. 
_V ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
76caovassg 6051 . . 3  |-  ( ( T.  /\  ( A  e.  _V  /\  B  e.  _V  /\  C  e. 
_V ) )  -> 
( ( A F B ) F C )  =  ( A F ( B F C ) ) )
84, 7mpan 424 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A F B ) F C )  =  ( A F ( B F C ) ) )
91, 2, 3, 8mp3an 1348 1  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    /\ w3a 980    = wceq 1364   T. wtru 1365    e. wcel 2160   _Vcvv 2752  (class class class)co 5892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5193  df-fv 5240  df-ov 5895
This theorem is referenced by:  caov32  6080  caov12  6081  caov31  6082  caov13  6083
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