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Theorem caovcom 6244
Description: Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)
Hypotheses
Ref Expression
caovcom.1  |-  A  e. 
_V
caovcom.2  |-  B  e. 
_V
caovcom.3  |-  ( x F y )  =  ( y F x )
Assertion
Ref Expression
caovcom  |-  ( A F B )  =  ( B F A )
Distinct variable groups:    x, y, A    x, B, y    x, F, y

Proof of Theorem caovcom
StepHypRef Expression
1 caovcom.1 . 2  |-  A  e. 
_V
2 caovcom.2 . . 3  |-  B  e. 
_V
31, 2pm3.2i 442 . 2  |-  ( A  e.  _V  /\  B  e.  _V )
4 caovcom.3 . . . 4  |-  ( x F y )  =  ( y F x )
54a1i 11 . . 3  |-  ( ( A  e.  _V  /\  ( x  e.  _V  /\  y  e.  _V )
)  ->  ( x F y )  =  ( y F x ) )
65caovcomg 6242 . 2  |-  ( ( A  e.  _V  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( A F B )  =  ( B F A ) )
71, 3, 6mp2an 654 1  |-  ( A F B )  =  ( B F A )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956  (class class class)co 6081
This theorem is referenced by:  caovord2  6259  caov32  6274  caov12  6275  caov42  6280  caovdir  6281  caovmo  6284  ecopovsym  7006  ecopover  7008  genpcl  8885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084
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