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

Theorem caovcan 6024
Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)
Hypotheses
Ref Expression
caovcan.1  |-  C  e. 
_V
caovcan.2  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( ( x F y )  =  ( x F z )  ->  y  =  z ) )
Assertion
Ref Expression
caovcan  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( ( A F B )  =  ( A F C )  ->  B  =  C ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    x, F, y, z    x, S, y, z

Proof of Theorem caovcan
StepHypRef Expression
1 oveq1 5865 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
2 oveq1 5865 . . . 4  |-  ( x  =  A  ->  (
x F C )  =  ( A F C ) )
31, 2eqeq12d 2297 . . 3  |-  ( x  =  A  ->  (
( x F y )  =  ( x F C )  <->  ( A F y )  =  ( A F C ) ) )
43imbi1d 308 . 2  |-  ( x  =  A  ->  (
( ( x F y )  =  ( x F C )  ->  y  =  C )  <->  ( ( A F y )  =  ( A F C )  ->  y  =  C ) ) )
5 oveq2 5866 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
65eqeq1d 2291 . . 3  |-  ( y  =  B  ->  (
( A F y )  =  ( A F C )  <->  ( A F B )  =  ( A F C ) ) )
7 eqeq1 2289 . . 3  |-  ( y  =  B  ->  (
y  =  C  <->  B  =  C ) )
86, 7imbi12d 311 . 2  |-  ( y  =  B  ->  (
( ( A F y )  =  ( A F C )  ->  y  =  C )  <->  ( ( A F B )  =  ( A F C )  ->  B  =  C ) ) )
9 caovcan.1 . . 3  |-  C  e. 
_V
10 oveq2 5866 . . . . . 6  |-  ( z  =  C  ->  (
x F z )  =  ( x F C ) )
1110eqeq2d 2294 . . . . 5  |-  ( z  =  C  ->  (
( x F y )  =  ( x F z )  <->  ( x F y )  =  ( x F C ) ) )
12 eqeq2 2292 . . . . 5  |-  ( z  =  C  ->  (
y  =  z  <->  y  =  C ) )
1311, 12imbi12d 311 . . . 4  |-  ( z  =  C  ->  (
( ( x F y )  =  ( x F z )  ->  y  =  z )  <->  ( ( x F y )  =  ( x F C )  ->  y  =  C ) ) )
1413imbi2d 307 . . 3  |-  ( z  =  C  ->  (
( ( x  e.  S  /\  y  e.  S )  ->  (
( x F y )  =  ( x F z )  -> 
y  =  z ) )  <->  ( ( x  e.  S  /\  y  e.  S )  ->  (
( x F y )  =  ( x F C )  -> 
y  =  C ) ) ) )
15 caovcan.2 . . 3  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( ( x F y )  =  ( x F z )  ->  y  =  z ) )
169, 14, 15vtocl 2838 . 2  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( ( x F y )  =  ( x F C )  ->  y  =  C ) )
174, 8, 16vtocl2ga 2851 1  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( ( A F B )  =  ( A F C )  ->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788  (class class class)co 5858
This theorem is referenced by:  ecopovtrn  6761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
  Copyright terms: Public domain W3C validator