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Theorem inuni 3937
 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 3612 . . . . 5
21anbi1i 439 . . . 4
3 elin 3154 . . . 4
4 ancom 257 . . . . . . . 8
5 r19.41v 2483 . . . . . . . 8
64, 5bitr4i 180 . . . . . . 7
76exbii 1512 . . . . . 6
8 rexcom4 2594 . . . . . 6
97, 8bitr4i 180 . . . . 5
10 vex 2577 . . . . . . . . . 10
1110inex1 3919 . . . . . . . . 9
12 eleq2 2117 . . . . . . . . 9
1311, 12ceqsexv 2610 . . . . . . . 8
14 elin 3154 . . . . . . . 8
1513, 14bitri 177 . . . . . . 7
1615rexbii 2348 . . . . . 6
17 r19.41v 2483 . . . . . 6
1816, 17bitri 177 . . . . 5
199, 18bitri 177 . . . 4
202, 3, 193bitr4i 205 . . 3
21 eluniab 3620 . . 3
2220, 21bitr4i 180 . 2
2322eqriv 2053 1
 Colors of variables: wff set class Syntax hints:   wa 101   wceq 1259  wex 1397   wcel 1409  cab 2042  wrex 2324   cin 2944  cuni 3608 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  ax-sep 3903 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-rex 2329  df-v 2576  df-in 2952  df-uni 3609 This theorem is referenced by: (None)
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