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

Theorem caovcanrd 6242
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 6235 . . 3  |-  ( ph  ->  ( A F B )  =  ( B F A ) )
5 caovcand.4 . . . 4  |-  ( ph  ->  C  e.  S )
61, 2, 5caovcomd 6235 . . 3  |-  ( ph  ->  ( A F C )  =  ( C F A ) )
74, 6eqeq12d 2449 . 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 6241 . 2  |-  ( ph  ->  ( ( A F B )  =  ( A F C )  <-> 
B  =  C ) )
117, 10bitr3d 247 1  |-  ( ph  ->  ( ( B F A )  =  ( C F A )  <-> 
B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725  (class class class)co 6073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076
  Copyright terms: Public domain W3C validator