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Theorem caovcan 6253
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 6090 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
2 oveq1 6090 . . . 4  |-  ( x  =  A  ->  (
x F C )  =  ( A F C ) )
31, 2eqeq12d 2452 . . 3  |-  ( x  =  A  ->  (
( x F y )  =  ( x F C )  <->  ( A F y )  =  ( A F C ) ) )
43imbi1d 310 . 2  |-  ( x  =  A  ->  (
( ( x F y )  =  ( x F C )  ->  y  =  C )  <->  ( ( A F y )  =  ( A F C )  ->  y  =  C ) ) )
5 oveq2 6091 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
65eqeq1d 2446 . . 3  |-  ( y  =  B  ->  (
( A F y )  =  ( A F C )  <->  ( A F B )  =  ( A F C ) ) )
7 eqeq1 2444 . . 3  |-  ( y  =  B  ->  (
y  =  C  <->  B  =  C ) )
86, 7imbi12d 313 . 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 6091 . . . . . 6  |-  ( z  =  C  ->  (
x F z )  =  ( x F C ) )
1110eqeq2d 2449 . . . . 5  |-  ( z  =  C  ->  (
( x F y )  =  ( x F z )  <->  ( x F y )  =  ( x F C ) ) )
12 eqeq2 2447 . . . . 5  |-  ( z  =  C  ->  (
y  =  z  <->  y  =  C ) )
1311, 12imbi12d 313 . . . 4  |-  ( z  =  C  ->  (
( ( x F y )  =  ( x F z )  ->  y  =  z )  <->  ( ( x F y )  =  ( x F C )  ->  y  =  C ) ) )
1413imbi2d 309 . . 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 3008 . 2  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( ( x F y )  =  ( x F C )  ->  y  =  C ) )
174, 8, 16vtocl2ga 3021 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 360    = wceq 1653    e. wcel 1726   _Vcvv 2958  (class class class)co 6083
This theorem is referenced by:  ecopovtrn  7009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
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