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Theorem caovdir2d 5619
Description: Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovdir2d.1  S  S  S  G F  G F G
caovdir2d.2  S
caovdir2d.3  S
caovdir2d.4  C  S
caovdir2d.cl  S  S  F  S
caovdir2d.com  S  S  G  G
Assertion
Ref Expression
caovdir2d  F G C  G C F G C
Distinct variable groups:   ,,,   ,,,   , C,,   ,,,   , F,,   , G,,   , S,,

Proof of Theorem caovdir2d
StepHypRef Expression
1 caovdir2d.1 . . 3  S  S  S  G F  G F G
2 caovdir2d.4 . . 3  C  S
3 caovdir2d.2 . . 3  S
4 caovdir2d.3 . . 3  S
51, 2, 3, 4caovdid 5618 . 2  C G F  C G F C G
6 caovdir2d.com . . 3  S  S  G  G
7 caovdir2d.cl . . . 4  S  S  F  S
87, 3, 4caovcld 5596 . . 3  F  S
96, 8, 2caovcomd 5599 . 2  F G C  C G F
106, 3, 2caovcomd 5599 . . 3  G C  C G
116, 4, 2caovcomd 5599 . . 3  G C  C G
1210, 11oveq12d 5473 . 2  G C F G C  C G F C G
135, 9, 123eqtr4d 2079 1  F G C  G C F G C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   w3a 884   wceq 1242   wcel 1390  (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-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:  addcmpblnq  6351  ltanqg  6384  addcmpblnq0  6425  mulasssrg  6666  mulgt0sr  6684  mulextsr1lem  6686
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