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Theorem caovcan 5690
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 5544 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
2 oveq1 5544 . . . 4  |-  ( x  =  A  ->  (
x F C )  =  ( A F C ) )
31, 2eqeq12d 2096 . . 3  |-  ( x  =  A  ->  (
( x F y )  =  ( x F C )  <->  ( A F y )  =  ( A F C ) ) )
43imbi1d 229 . 2  |-  ( x  =  A  ->  (
( ( x F y )  =  ( x F C )  ->  y  =  C )  <->  ( ( A F y )  =  ( A F C )  ->  y  =  C ) ) )
5 oveq2 5545 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
65eqeq1d 2090 . . 3  |-  ( y  =  B  ->  (
( A F y )  =  ( A F C )  <->  ( A F B )  =  ( A F C ) ) )
7 eqeq1 2088 . . 3  |-  ( y  =  B  ->  (
y  =  C  <->  B  =  C ) )
86, 7imbi12d 232 . 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 5545 . . . . . 6  |-  ( z  =  C  ->  (
x F z )  =  ( x F C ) )
1110eqeq2d 2093 . . . . 5  |-  ( z  =  C  ->  (
( x F y )  =  ( x F z )  <->  ( x F y )  =  ( x F C ) ) )
12 eqeq2 2091 . . . . 5  |-  ( z  =  C  ->  (
y  =  z  <->  y  =  C ) )
1311, 12imbi12d 232 . . . 4  |-  ( z  =  C  ->  (
( ( x F y )  =  ( x F z )  ->  y  =  z )  <->  ( ( x F y )  =  ( x F C )  ->  y  =  C ) ) )
1413imbi2d 228 . . 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 2654 . 2  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( ( x F y )  =  ( x F C )  ->  y  =  C ) )
174, 8, 16vtocl2ga 2667 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 102    = wceq 1285    e. wcel 1434   _Vcvv 2602  (class class class)co 5537
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-rex 2355  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-iota 4891  df-fv 4934  df-ov 5540
This theorem is referenced by: (None)
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