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Theorem caovcom 6059
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 441 . 2  |-  ( A  e.  _V  /\  B  e.  _V )
4 caovcom.3 . . . 4  |-  ( x F y )  =  ( y F x )
54a1i 10 . . 3  |-  ( ( A  e.  _V  /\  ( x  e.  _V  /\  y  e.  _V )
)  ->  ( x F y )  =  ( y F x ) )
65caovcomg 6057 . 2  |-  ( ( A  e.  _V  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( A F B )  =  ( B F A ) )
71, 3, 6mp2an 653 1  |-  ( A F B )  =  ( B F A )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822  (class class class)co 5900
This theorem is referenced by:  caovord2  6074  caov32  6089  caov12  6090  caov42  6095  caovdir  6096  caovmo  6099  ecopovsym  6803  ecopover  6805  genpcl  8677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-ov 5903
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