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Theorem coundi 4850
 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 3864 . . 3
2 brun 3838 . . . . . . . 8
32anbi1i 439 . . . . . . 7
4 andir 743 . . . . . . 7
53, 4bitri 177 . . . . . 6
65exbii 1512 . . . . 5
7 19.43 1535 . . . . 5
86, 7bitr2i 178 . . . 4
98opabbii 3852 . . 3
101, 9eqtri 2076 . 2
11 df-co 4382 . . 3
12 df-co 4382 . . 3
1311, 12uneq12i 3123 . 2
14 df-co 4382 . 2
1510, 13, 143eqtr4ri 2087 1
 Colors of variables: wff set class Syntax hints:   wa 101   wo 639   wceq 1259  wex 1397   cun 2943   class class class wbr 3792  copab 3845   ccom 4377 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-br 3793  df-opab 3847  df-co 4382 This theorem is referenced by:  relcoi1  4877
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