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Theorem caovcang 5604
Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypothesis
Ref Expression
caovcang.1  T  S  S  F  F
Assertion
Ref Expression
caovcang  T  S  C  S  F  F C  C
Distinct variable groups:   ,,,   ,,,   , C,,   ,,,   , F,,   , S,,   , T,,

Proof of Theorem caovcang
StepHypRef Expression
1 caovcang.1 . . 3  T  S  S  F  F
21ralrimivvva 2396 . 2  T  S  S  F  F
3 oveq1 5462 . . . . 5  F  F
4 oveq1 5462 . . . . 5  F  F
53, 4eqeq12d 2051 . . . 4  F  F  F  F
65bibi1d 222 . . 3  F  F  F  F
7 oveq2 5463 . . . . 5  F  F
87eqeq1d 2045 . . . 4  F  F  F  F
9 eqeq1 2043 . . . 4
108, 9bibi12d 224 . . 3  F  F  F  F
11 oveq2 5463 . . . . 5  C  F  F C
1211eqeq2d 2048 . . . 4  C  F  F  F  F C
13 eqeq2 2046 . . . 4  C  C
1412, 13bibi12d 224 . . 3  C  F  F  F  F C  C
156, 10, 14rspc3v 2659 . 2  T  S  C  S  T  S  S  F  F  F  F C  C
162, 15mpan9 265 1  T  S  C  S  F  F C  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wcel 1390  wral 2300  (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-bndl 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-ral 2305  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:  caovcand  5605
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