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Theorem uniin 3600
 Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
uniin

Proof of Theorem uniin
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.40 1522 . . . 4
2 elin 3126 . . . . . . 7
32anbi2i 430 . . . . . 6
4 anandi 524 . . . . . 6
53, 4bitri 173 . . . . 5
65exbii 1496 . . . 4
7 eluni 3583 . . . . 5
8 eluni 3583 . . . . 5
97, 8anbi12i 433 . . . 4
101, 6, 93imtr4i 190 . . 3
11 eluni 3583 . . 3
12 elin 3126 . . 3
1310, 11, 123imtr4i 190 . 2
1413ssriv 2949 1
 Colors of variables: wff set class Syntax hints:   wa 97  wex 1381   wcel 1393   cin 2916   wss 2917  cuni 3580 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581 This theorem is referenced by: (None)
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