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Theorem caovcomg 6015
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 2635 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x F y )  =  ( y F x ) )
3 oveq1 5865 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
4 oveq2 5866 . . . 4  |-  ( x  =  A  ->  (
y F x )  =  ( y F A ) )
53, 4eqeq12d 2297 . . 3  |-  ( x  =  A  ->  (
( x F y )  =  ( y F x )  <->  ( A F y )  =  ( y F A ) ) )
6 oveq2 5866 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
7 oveq1 5865 . . . 4  |-  ( y  =  B  ->  (
y F A )  =  ( B F A ) )
86, 7eqeq12d 2297 . . 3  |-  ( y  =  B  ->  (
( A F y )  =  ( y F A )  <->  ( A F B )  =  ( B F A ) ) )
95, 8rspc2v 2890 . 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 455 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 358    = wceq 1623    e. wcel 1684   A.wral 2543  (class class class)co 5858
This theorem is referenced by:  caovcomd  6016  caovcom  6017  caofcom  6109  seqcaopr  11083  cmncom  15105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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