ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caovcomg Unicode version

Theorem caovcomg 5681
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 2444 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x F y )  =  ( y F x ) )
3 oveq1 5544 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
4 oveq2 5545 . . . 4  |-  ( x  =  A  ->  (
y F x )  =  ( y F A ) )
53, 4eqeq12d 2096 . . 3  |-  ( x  =  A  ->  (
( x F y )  =  ( y F x )  <->  ( A F y )  =  ( y F A ) ) )
6 oveq2 5545 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
7 oveq1 5544 . . . 4  |-  ( y  =  B  ->  (
y F A )  =  ( B F A ) )
86, 7eqeq12d 2096 . . 3  |-  ( y  =  B  ->  (
( A F y )  =  ( y F A )  <->  ( A F B )  =  ( B F A ) ) )
95, 8rspc2v 2714 . 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 275 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 102    = wceq 1285    e. wcel 1434   A.wral 2349  (class class class)co 5537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-iota 4891  df-fv 4934  df-ov 5540
This theorem is referenced by:  caovcomd  5682  caovcom  5683  caovlem2d  5718  caofcom  5759  iseqcaopr  9547
  Copyright terms: Public domain W3C validator