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Theorem iunin1f 28173
 Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4352 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) (Revised by Thierry Arnoux, 2-May-2020.)
Hypothesis
Ref Expression
iunin1f.1
Assertion
Ref Expression
iunin1f

Proof of Theorem iunin1f
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfcv 2580 . . . . . 6
2 iunin1f.1 . . . . . 6
31, 2nfel 2593 . . . . 5
43r19.41 2978 . . . 4
5 elin 3649 . . . . 5
65rexbii 2924 . . . 4
7 eliun 4304 . . . . 5
87anbi1i 699 . . . 4
94, 6, 83bitr4i 280 . . 3
10 eliun 4304 . . 3
11 elin 3649 . . 3
129, 10, 113bitr4i 280 . 2
1312eqriv 2418 1
 Colors of variables: wff setvar class Syntax hints:   wa 370   wceq 1437   wcel 1872  wnfc 2566  wrex 2772   cin 3435  ciun 4299 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401 This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082  df-in 3443  df-iun 4301 This theorem is referenced by:  esum2dlem  28921
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