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

Theorem caovcanrd 6005
Description: Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovcang.1  |-  ( (
ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y )  =  ( x F z )  <-> 
y  =  z ) )
caovcand.2  |-  ( ph  ->  A  e.  T )
caovcand.3  |-  ( ph  ->  B  e.  S )
caovcand.4  |-  ( ph  ->  C  e.  S )
caovcanrd.5  |-  ( ph  ->  A  e.  S )
caovcanrd.6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
Assertion
Ref Expression
caovcanrd  |-  ( ph  ->  ( ( B F A )  =  ( C F A )  <-> 
B  =  C ) )
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    ph, x, y, z   
x, F, y, z   
x, S, y, z   
x, T, y, z

Proof of Theorem caovcanrd
StepHypRef Expression
1 caovcanrd.6 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
2 caovcanrd.5 . . . 4  |-  ( ph  ->  A  e.  S )
3 caovcand.3 . . . 4  |-  ( ph  ->  B  e.  S )
41, 2, 3caovcomd 5998 . . 3  |-  ( ph  ->  ( A F B )  =  ( B F A ) )
5 caovcand.4 . . . 4  |-  ( ph  ->  C  e.  S )
61, 2, 5caovcomd 5998 . . 3  |-  ( ph  ->  ( A F C )  =  ( C F A ) )
74, 6eqeq12d 2180 . 2  |-  ( ph  ->  ( ( A F B )  =  ( A F C )  <-> 
( B F A )  =  ( C F A ) ) )
8 caovcang.1 . . 3  |-  ( (
ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y )  =  ( x F z )  <-> 
y  =  z ) )
9 caovcand.2 . . 3  |-  ( ph  ->  A  e.  T )
108, 9, 3, 5caovcand 6004 . 2  |-  ( ph  ->  ( ( A F B )  =  ( A F C )  <-> 
B  =  C ) )
117, 10bitr3d 189 1  |-  ( ph  ->  ( ( B F A )  =  ( C F A )  <-> 
B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 968    = wceq 1343    e. wcel 2136  (class class class)co 5842
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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator