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Theorem caovcomd 6235
Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovcomg.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
caovcomd.2  |-  ( ph  ->  A  e.  S )
caovcomd.3  |-  ( ph  ->  B  e.  S )
Assertion
Ref Expression
caovcomd  |-  ( ph  ->  ( A F B )  =  ( B F A ) )
Distinct variable groups:    x, y, A    x, B, y    ph, x, y    x, F, y    x, S, y

Proof of Theorem caovcomd
StepHypRef Expression
1 id 20 . 2  |-  ( ph  ->  ph )
2 caovcomd.2 . 2  |-  ( ph  ->  A  e.  S )
3 caovcomd.3 . 2  |-  ( ph  ->  B  e.  S )
4 caovcomg.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
54caovcomg 6234 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A F B )  =  ( B F A ) )
61, 2, 3, 5syl12anc 1182 1  |-  ( ph  ->  ( A F B )  =  ( B F A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725  (class class class)co 6073
This theorem is referenced by:  caovcanrd  6242  caovord2d  6248  caovdir2d  6255  caov32d  6259  caov12d  6260  caov31d  6261  caov411d  6264  caov42d  6265  seqf1olem2a  11353
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076
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