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Theorem caovcan 6267
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 6098 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
2 oveq1 6098 . . . 4  |-  ( x  =  A  ->  (
x F C )  =  ( A F C ) )
31, 2eqeq12d 2457 . . 3  |-  ( x  =  A  ->  (
( x F y )  =  ( x F C )  <->  ( A F y )  =  ( A F C ) ) )
43imbi1d 317 . 2  |-  ( x  =  A  ->  (
( ( x F y )  =  ( x F C )  ->  y  =  C )  <->  ( ( A F y )  =  ( A F C )  ->  y  =  C ) ) )
5 oveq2 6099 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
65eqeq1d 2451 . . 3  |-  ( y  =  B  ->  (
( A F y )  =  ( A F C )  <->  ( A F B )  =  ( A F C ) ) )
7 eqeq1 2449 . . 3  |-  ( y  =  B  ->  (
y  =  C  <->  B  =  C ) )
86, 7imbi12d 320 . 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 6099 . . . . . 6  |-  ( z  =  C  ->  (
x F z )  =  ( x F C ) )
1110eqeq2d 2454 . . . . 5  |-  ( z  =  C  ->  (
( x F y )  =  ( x F z )  <->  ( x F y )  =  ( x F C ) ) )
12 eqeq2 2452 . . . . 5  |-  ( z  =  C  ->  (
y  =  z  <->  y  =  C ) )
1311, 12imbi12d 320 . . . 4  |-  ( z  =  C  ->  (
( ( x F y )  =  ( x F z )  ->  y  =  z )  <->  ( ( x F y )  =  ( x F C )  ->  y  =  C ) ) )
1413imbi2d 316 . . 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 3024 . 2  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( ( x F y )  =  ( x F C )  ->  y  =  C ) )
174, 8, 16vtocl2ga 3038 1  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( ( A F B )  =  ( A F C )  ->  B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972  (class class class)co 6091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-ov 6094
This theorem is referenced by:  ecopovtrn  7203
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