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Theorem caovass 5981
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 1339 . . 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 5979 . . 3  |-  ( ( T.  /\  ( A  e.  _V  /\  B  e.  _V  /\  C  e. 
_V ) )  -> 
( ( A F B ) F C )  =  ( A F ( B F C ) ) )
84, 7mpan 421 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A F B ) F C )  =  ( A F ( B F C ) ) )
91, 2, 3, 8mp3an 1319 1  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    /\ w3a 963    = wceq 1335   T. wtru 1336    e. wcel 2128   _Vcvv 2712  (class class class)co 5824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-iota 5135  df-fv 5178  df-ov 5827
This theorem is referenced by:  caov32  6008  caov12  6009  caov31  6010  caov13  6011
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