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Theorem iunxun 4047
 Description: Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iunxun x (AB)C = (x A Cx B C)

Proof of Theorem iunxun
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 rexun 3443 . . . 4 (x (AB)y C ↔ (x A y C x B y C))
2 eliun 3973 . . . . 5 (y x A Cx A y C)
3 eliun 3973 . . . . 5 (y x B Cx B y C)
42, 3orbi12i 507 . . . 4 ((y x A C y x B C) ↔ (x A y C x B y C))
51, 4bitr4i 243 . . 3 (x (AB)y C ↔ (y x A C y x B C))
6 eliun 3973 . . 3 (y x (AB)Cx (AB)y C)
7 elun 3220 . . 3 (y (x A Cx B C) ↔ (y x A C y x B C))
85, 6, 73bitr4i 268 . 2 (y x (AB)Cy (x A Cx B C))
98eqriv 2350 1 x (AB)C = (x A Cx B C)
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∪ cun 3207  ∪ciun 3969 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-iun 3971 This theorem is referenced by: (None)
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