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Theorem caovass 6360
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 1374 . . 3  |- T.
5 caovass.4 . . . . 5  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
65a1i 11 . . . 4  |-  ( ( T.  /\  ( x  e.  _V  /\  y  e.  _V  /\  z  e. 
_V ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
76caovassg 6358 . . 3  |-  ( ( T.  /\  ( A  e.  _V  /\  B  e.  _V  /\  C  e. 
_V ) )  -> 
( ( A F B ) F C )  =  ( A F ( B F C ) ) )
84, 7mpan 670 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A F B ) F C )  =  ( A F ( B F C ) ) )
91, 2, 3, 8mp3an 1315 1  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 965    = wceq 1370   T. wtru 1371    e. wcel 1758   _Vcvv 3065  (class class class)co 6187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-iota 5476  df-fv 5521  df-ov 6190
This theorem is referenced by:  caov32  6387  caov12  6388  caov31  6389  caov13  6390  caov4  6391  caov411  6392  caovdilem  6395  caovmo  6397
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