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Theorem caovass 6249
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 1331 . . 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 6247 . . 3  |-  ( (  T.  /\  ( A  e.  _V  /\  B  e.  _V  /\  C  e. 
_V ) )  -> 
( ( A F B ) F C )  =  ( A F ( B F C ) ) )
84, 7mpan 653 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A F B ) F C )  =  ( A F ( B F C ) ) )
91, 2, 3, 8mp3an 1280 1  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    /\ w3a 937    T. wtru 1326    = wceq 1653    e. wcel 1726   _Vcvv 2958  (class class class)co 6083
This theorem is referenced by:  caov32  6276  caov12  6277  caov31  6278  caov13  6279  caov4  6280  caov411  6281  caovdilem  6284  caovmo  6286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
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