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Theorem inuni 4614
 Description: The intersection of a union with a class is equal to the union of the intersections of each element of with . (Contributed by FL, 24-Mar-2007.)
Assertion
Ref Expression
inuni
Distinct variable groups:   ,,   ,,

Proof of Theorem inuni
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eluni2 4253 . . . . 5
21anbi1i 695 . . . 4
3 elin 3686 . . . 4
4 ancom 450 . . . . . . . 8
5 r19.41v 3009 . . . . . . . 8
64, 5bitr4i 252 . . . . . . 7
76exbii 1667 . . . . . 6
8 rexcom4 3129 . . . . . 6
97, 8bitr4i 252 . . . . 5
10 vex 3112 . . . . . . . . . 10
1110inex1 4593 . . . . . . . . 9
12 eleq2 2530 . . . . . . . . 9
1311, 12ceqsexv 3146 . . . . . . . 8
14 elin 3686 . . . . . . . 8
1513, 14bitri 249 . . . . . . 7
1615rexbii 2959 . . . . . 6
17 r19.41v 3009 . . . . . 6
1816, 17bitri 249 . . . . 5
199, 18bitri 249 . . . 4
202, 3, 193bitr4i 277 . . 3
21 eluniab 4260 . . 3
2220, 21bitr4i 252 . 2
2322eqriv 2453 1
 Colors of variables: wff setvar class Syntax hints:  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  E.wrex 2808  i^icin 3474  U.cuni 4249 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-uni 4250
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