MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caovcanrd Structured version   Unicode version

Theorem caovcanrd 6477
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 6470 . . 3  |-  ( ph  ->  ( A F B )  =  ( B F A ) )
5 caovcand.4 . . . 4  |-  ( ph  ->  C  e.  S )
61, 2, 5caovcomd 6470 . . 3  |-  ( ph  ->  ( A F C )  =  ( C F A ) )
74, 6eqeq12d 2479 . 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 6476 . 2  |-  ( ph  ->  ( ( A F B )  =  ( A F C )  <-> 
B  =  C ) )
117, 10bitr3d 255 1  |-  ( ph  ->  ( ( B F A )  =  ( C F A )  <-> 
B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819  (class class class)co 6296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator