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Theorem caovcomd 6183
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 6182 . 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 1649    e. wcel 1717  (class class class)co 6021
This theorem is referenced by:  caovcanrd  6190  caovord2d  6196  caovdir2d  6203  caov32d  6207  caov12d  6208  caov31d  6209  caov411d  6212  caov42d  6213  seqf1olem2a  11289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-ov 6024
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