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Theorem inuni 4364
 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 4021 . . . . 5
21anbi1i 678 . . . 4
3 elin 3532 . . . 4
4 ancom 439 . . . . . . . 8
5 r19.41v 2863 . . . . . . . 8
64, 5bitr4i 245 . . . . . . 7
76exbii 1593 . . . . . 6
8 rexcom4 2977 . . . . . 6
97, 8bitr4i 245 . . . . 5
10 vex 2961 . . . . . . . . . 10
1110inex1 4346 . . . . . . . . 9
12 eleq2 2499 . . . . . . . . 9
1311, 12ceqsexv 2993 . . . . . . . 8
14 elin 3532 . . . . . . . 8
1513, 14bitri 242 . . . . . . 7
1615rexbii 2732 . . . . . 6
17 r19.41v 2863 . . . . . 6
1816, 17bitri 242 . . . . 5
199, 18bitri 242 . . . 4
202, 3, 193bitr4i 270 . . 3
21 eluniab 4029 . . 3
2220, 21bitr4i 245 . 2
2322eqriv 2435 1
 Colors of variables: wff set class Syntax hints:   wa 360  wex 1551   wceq 1653   wcel 1726  cab 2424  wrex 2708   cin 3321  cuni 4017 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-v 2960  df-in 3329  df-uni 4018
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