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Theorem caovcan 5607
Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)
Hypotheses
Ref Expression
caovcan.1  C 
_V
caovcan.2  S  S  F  F
Assertion
Ref Expression
caovcan  S  S  F  F C  C
Distinct variable groups:   ,,,   ,,,   , C,,   , F,,   , S,,

Proof of Theorem caovcan
StepHypRef Expression
1 oveq1 5462 . . . 4  F  F
2 oveq1 5462 . . . 4  F C  F C
31, 2eqeq12d 2051 . . 3  F  F C  F  F C
43imbi1d 220 . 2  F  F C  C  F  F C  C
5 oveq2 5463 . . . 4  F  F
65eqeq1d 2045 . . 3  F  F C  F  F C
7 eqeq1 2043 . . 3  C  C
86, 7imbi12d 223 . 2  F  F C  C  F  F C  C
9 caovcan.1 . . 3  C 
_V
10 oveq2 5463 . . . . . 6  C  F  F C
1110eqeq2d 2048 . . . . 5  C  F  F  F  F C
12 eqeq2 2046 . . . . 5  C  C
1311, 12imbi12d 223 . . . 4  C  F  F  F  F C  C
1413imbi2d 219 . . 3  C  S  S  F  F  S  S  F  F C  C
15 caovcan.2 . . 3  S  S  F  F
169, 14, 15vtocl 2602 . 2  S  S  F  F C  C
174, 8, 16vtocl2ga 2615 1  S  S  F  F C  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390   _Vcvv 2551  (class class class)co 5455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by: (None)
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