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Theorem coundi 4765
 Description: Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
coundi

Proof of Theorem coundi
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unopab 3827 . . 3
2 brun 3801 . . . . . . . 8
32anbi1i 431 . . . . . . 7
4 andir 731 . . . . . . 7
53, 4bitri 173 . . . . . 6
65exbii 1493 . . . . 5
7 19.43 1516 . . . . 5
86, 7bitr2i 174 . . . 4
98opabbii 3815 . . 3
101, 9eqtri 2057 . 2
11 df-co 4297 . . 3
12 df-co 4297 . . 3
1311, 12uneq12i 3089 . 2
14 df-co 4297 . 2
1510, 13, 143eqtr4ri 2068 1
 Colors of variables: wff set class Syntax hints:   wa 97   wo 628   wceq 1242  wex 1378   cun 2909   class class class wbr 3755  copab 3808   ccom 4292 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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-br 3756  df-opab 3810  df-co 4297 This theorem is referenced by:  relcoi1  4792
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