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Theorem caovcomg 5919
Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)
Hypothesis
Ref Expression
caovcomg.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
Assertion
Ref Expression
caovcomg  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( 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 caovcomg
StepHypRef Expression
1 caovcomg.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
21ralrimivva 2512 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x F y )  =  ( y F x ) )
3 oveq1 5774 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
4 oveq2 5775 . . . 4  |-  ( x  =  A  ->  (
y F x )  =  ( y F A ) )
53, 4eqeq12d 2152 . . 3  |-  ( x  =  A  ->  (
( x F y )  =  ( y F x )  <->  ( A F y )  =  ( y F A ) ) )
6 oveq2 5775 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
7 oveq1 5774 . . . 4  |-  ( y  =  B  ->  (
y F A )  =  ( B F A ) )
86, 7eqeq12d 2152 . . 3  |-  ( y  =  B  ->  (
( A F y )  =  ( y F A )  <->  ( A F B )  =  ( B F A ) ) )
95, 8rspc2v 2797 . 2  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A. x  e.  S  A. y  e.  S  ( x F y )  =  ( y F x )  ->  ( A F B )  =  ( B F A ) ) )
102, 9mpan9 279 1  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A F B )  =  ( B F A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2414  (class class class)co 5767
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 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770
This theorem is referenced by:  caovcomd  5920  caovcom  5921  caovlem2d  5956  caofcom  5998  seq3caopr  10249
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