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Theorem uniin 3724
 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 1593 . . . 4
2 elin 3227 . . . . . . 7
32anbi2i 450 . . . . . 6
4 anandi 562 . . . . . 6
53, 4bitri 183 . . . . 5
65exbii 1567 . . . 4
7 eluni 3707 . . . . 5
8 eluni 3707 . . . . 5
97, 8anbi12i 453 . . . 4
101, 6, 93imtr4i 200 . . 3
11 eluni 3707 . . 3
12 elin 3227 . . 3
1310, 11, 123imtr4i 200 . 2
1413ssriv 3069 1
 Colors of variables: wff set class Syntax hints:   wa 103  wex 1451   wcel 1463   cin 3038   wss 3039  cuni 3704 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-uni 3705 This theorem is referenced by:  tgval  12113
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