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Theorem coundir 4853
 Description: Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
coundir

Proof of Theorem coundir
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unopab 3865 . . 3
2 brun 3839 . . . . . . . 8
32anbi2i 445 . . . . . . 7
4 andi 765 . . . . . . 7
53, 4bitri 182 . . . . . 6
65exbii 1537 . . . . 5
7 19.43 1560 . . . . 5
86, 7bitr2i 183 . . . 4
98opabbii 3853 . . 3
101, 9eqtri 2102 . 2
11 df-co 4380 . . 3
12 df-co 4380 . . 3
1311, 12uneq12i 3125 . 2
14 df-co 4380 . 2
1510, 13, 143eqtr4ri 2113 1
 Colors of variables: wff set class Syntax hints:   wa 102   wo 662   wceq 1285  wex 1422   cun 2972   class class class wbr 3793  copab 3846   ccom 4375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-br 3794  df-opab 3848  df-co 4380 This theorem is referenced by: (None)
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