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

Theorem caovcomd 5688
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 5687 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A F B )  =  ( B F A ) )
61, 2, 3, 5syl12anc 1168 1  |-  ( ph  ->  ( A F B )  =  ( B F A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434  (class class class)co 5543
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 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546
This theorem is referenced by:  caovcanrd  5695  caovord2d  5701  caovdir2d  5708  caov32d  5712  caov12d  5713  caov31d  5714  caov411d  5717  caov42d  5718  caovimo  5725  ecopovsymg  6271  ecopoverg  6273  ltsonq  6650  prarloclemlo  6746  addextpr  6873  ltsosr  7003  ltasrg  7009  mulextsr1lem  7018
  Copyright terms: Public domain W3C validator