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Theorem caovcomd 5927
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 19 . 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 5926 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A F B )  =  ( B F A ) )
61, 2, 3, 5syl12anc 1214 1  |-  ( ph  ->  ( A F B )  =  ( B F A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480  (class class class)co 5774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  caovcanrd  5934  caovord2d  5940  caovdir2d  5947  caov32d  5951  caov12d  5952  caov31d  5953  caov411d  5956  caov42d  5957  caovimo  5964  ecopovsymg  6528  ecopoverg  6530  ltsonq  7213  prarloclemlo  7309  addextpr  7436  ltsosr  7579  ltasrg  7585  mulextsr1lem  7595  seq3f1olemqsumkj  10278
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